Ulrich Krengel
Ergodic Theorems
With a Supplement by Antoine Brunei
w
DE
G
Walter de Gruyter
Berlin · New York 1985

Author
Dr. Ulrich Krengel
Professor of Mathematics
Universitat Gottingen
Library of Congress Cataloging in Publication Data
Krengel, Ulrich, 1937-
Ergodic theorems.
(De Gruyter studies
Bibliography: p.
Includes index.
1. Ergodic theory.
III. Series.
QA313.K74 1985
in mathematics ; 6)
I. Brunei,
515.4'2
ISBN 0-89925-024-6 (U.S.)
Antoine.
II.
85-4457
Title.
CIP-Kurztitelaufnahme der Deutschen Bibliothek
Krengel, Ulrich:
Ergodic theorems / Ulrich Krengel. With a suppl. by Antoine Brunei.
- Berlin ; New York : de Gruyter 1985.
(De Gruyter studies in mathematics ; 6)
ISBN 3-11-008478-3
NE:GT
© Copyright 1985 by Walter de Gruyter & Co., Berlin. All rights reserved, including those of
translation into foreign languages. No part of this book may be reproduced in any form - by photoprint,
microfilm, or any other means - nor transmitted nor translated into a machine language without
written permission from the publisher. Printed in Germany.
Cover design: Rudolf Hubler, Berlin. Typesetting and Printing: Tutte Druckerei GmbH, Salzweg-
Passau. Binding: Luderitz & Bauer, Berlin.

Lesen heiBt borgen,
daraus erfinden abtragen.
G. C. Lichtenberg,
Sudelbuch F
Preface
The study of ergodic theorems is the oldest branch of ergodic theory. It was
started in 1931 by von Neumann and Birkhoff, having its origins in statistical
mechanics. While new applications to mathematical physics continued to come
in, the theory soon earned its own rights as an important chapter in functional
analysis and probability.
So far, a comprehensive treatment has been neglected, and this book tries to
provide it. Most of its material has not appeared in any other book. This applies
even to older results, but the main body of the results is less than twenty years old
and several interesting topics have just been added in the last decade.
Roughly speaking we ask: When do averages of quantities generated in a
stationary manner converge? In the classical situation the stationarity is
described by a measure preserving transformation τ, and one considers averages
taken along a sequence/,/<>τ,/°τ2,... for integrable/. This corresponds to the
probabilistic concept of stationarity. More generally, τ can be replaced by an
operator Tin a function space and/° τ1' by Tlf. As Tl is the result of the iterated
action of the same operator we again have some kind of stationarity.
Generalizing further, we study semigroups {Tg, g e G} of operators and limits of averages
of Tgfovev subsets In с G. The term "ergodic theorem" has been used by some
authors for quite distinct limit theorems, but we reserve it for convergence
theorems dealing with such averages and for their close relatives. This meaning
seems most widely accepted. Among the relatives we count subadditive ergodic
theorems, local ergodic theorems (generalizing the differentiation of integrals),
ratio ergodic theorems and ergodic theorems for information.
The modes of convergence under consideration mostly are norm convergence
for "mean" ergodic theorems, and convergence almost everywhere for
"individual" (or "pointwise") ergodic theorems. Recently, weak convergence has gained
importance for nonlinear ergodic theorems and almost uniform convergence for
ergodic theorems in von Neumann algebras. Convergence in distribution will not
be considered. Typically, it applies to renormed averages rather than to averages,
and it requires different tools.
I have tried to make the various parts of the book independently readable. The
reader should start with any section which is of interest to him. He will then
notice which previous results enter and find that often just a few will suffice. Of

VI Preface
course, this implies some redundancy. On the other hand, I hope that this way
large portions of the book can serve as a textbook, and that this approach will
render this monograph useful and readily accessible for non-specialists.
I have not always given the shortest proof. Sometimes a longer proof seemed
more transparent. Another aspect has been the wish to introduce a variety of
methods. In some "additive" ergodic theorems the proof of convergence could
have been simplified by the use of subadditive theorems. However, the longer
additive arguments give access to an evaluation of the limit.
I presuppose knowledge of basic measure theory, and, for many sections, some
functional analysis. But I tried to help the non-experts with references even for
standard theorems.
Surely this book is biased towards my personal interests and even more so
since I have included a number of new results and proofs. But I also tried hard not
to miss any important result and to give it fair coverage. If a good presentation
existed I sometimes may have just quoted it. I apologize in advance to anyone
whose contributions were overlooked.
Concerning convergence almost everywhere the book of Stout [1974] covers
many of the themes in the complement of this book.
Most sections end with Notes containing additional information. But I did
include credits in the main text when it seemed possible without much delay.
Theorems, lemmas, definitions etc. are numbered consecutively in each
section. A quotation "theorem 2.3.4" refers to chapter 2, section 3, theorem 3.4. A
quotation "theorem 3.4" refers to the current chapter.
My indebtedness extends to many. First, I would like to thank my teacher,
Konrad Jacobs, for generating my interest in ergodic theory, for giving me a
sound introduction, and for suggesting a fertile area of research. I also owe much
to Louis Sucheston and his infectious enthusiasm. I am very grateful to M. Lin,
A. del Junco, Y. Derriennic, G. Keller, M. Akcoglu, R. Nagel, M. Mathieu, H.-J.
Borchers, A. Bellow, R. Jajte, W. Takahashi, J. Fritz, M. Keane, M. Denker and
many others for useful hints and encouragement. Martina Hochhaus helped with
the bibliography. Marrie Powell contributed her unusual skill at mathematical
typing. Special thanks go to Heinz Bauer for inviting the book into this series.
The idea for this book arose in 1976 at the end of a pleasant and interesting
sabbatical I spent at the University of Paris VI. Antoine Brunei and I agreed that
a book covering the whole spectrum of ergodic theorems was badly needed and
planned to write it jointly. Unfortunately, grave personal reasons prevented this.
I am the more grateful to Antoine Brunei for writing the supplement on Harris
processes, a topic to which he contributed so much.
I devote this book to my wife Beate, and to Jeannette Brunei who died of
cancer in 1981. They provided the environment for us in which devotion to
mathematical work was possible.
Gottingen, April 1985
Ulrich Krengel

Contents
Chapter 1: Measure preserving and null preserving point mappings 1
§1.1 Von Neumann's mean ergodic theorem, ergodicity 1
§1.2 BirkhofFs ergodic theorem 7
§1.3 Recurrence 16
§1.4 Shift transformations and stationary processes 22
§1.5 Kingman's subadditive ergodic theorem and
the multiplicative ergodic theorem of Oseledec 35
§1.6 Relatives of the maximal ergodic theorem 50
§1.7 Some general tools and principles 63
Chapter 2: Mean ergodic theory 71
§ 2.1 The mean ergodic theorem 71
§ 2.2 Uniform convergence 86
§2.3 Weak mixing, continuous spectrum and multiple recurrence 94
§ 2.4 The splitting theorem of Jacobs-Deleeuw-Glicksberg 103
Chapter 3: Positive contractions in Lx 113
§ 3.1 The Hopf decomposition 113
§ 3.2 The Chacon-Ornstein theorem 119
§ 3.3 Brunei's lemma and the identification of the limit 123
§ 3.4 Existence of finite invariant measures 135
§ 3.5 The subadditive ergodic theorem for positive contractions in Lx .. 146
§3.6 An example with divergence of Cesar о averages 151
§ 3.7 More on the filling scheme 154
Chapter 4: Extensions of the Lx-theory 159
§ 4.1 Non positive contractions in Lx · 159
§ 4.2 Vector valued ergodic theorems 167
§4.3 Power bounded operators and harmonic functions 172

VIII Contents
Chapter 5: Operators in C{K) and in Lp? (1 <p < oo) 177
§ 5.1 Markov operators in C(K) 177
§ 5.2 Contractions in Lp? (1 <p < oo) 186
Chapter 6: Pointwise ergodic theorems for multiparameter and amenable
semigroups 195
§6.1 Unrestricted convergence for averages over d-dimensional
intervals 195
§ 6.2 Multiparameter additive and subadditive processes 201
§ 6.3 Multiparameter semigroups of Lx-contractions 211
§ 6.4 Amenable semigroups 221
Chapter 7: Local ergodic theorems and differentiation 229
§ 7.1 Positive 1-parameter semigroups 229
§ 7.2 Local ergodic theorems for multiparameter and non positive
semigroups, and for vector valued functions 243
Chapter 8: Subsequences and generalized means 251
§ 8.1 Strong convergence and mixing 251
§ 8.2 Pointwise convergence 257
Chapter 9: Special topics 267
§9.1 Ergodic theorems in von Neumann algebras 267
§ 9.2 Entropy and information 281
§ 9.3 Nonlinear nonexpansive mappings 288
§ 9.4 Miscellanea 297
Supplement: Harris Processes, Special Functions, Zero-Two-Law
(by Antoine Brunei) 301
Bibliography 321
Notation 347
Index 351

Chapter 1: Measure preserving and null
preserving point mappings
We begin with the classical ergodic theorems for measure preserving
transformations and with their role in the theory of stationary processes. Then the
subadditive ergodic theorem is proved. We use it to derive the multiplicative ergodic
theorem of Oseledec, a powerful tool in the study of dynamical systems.
Recurrence is discussed for the wider class of null preserving transformations. The
dominated ergodic theorem provides important estimates of the supremum of
averages. Some topics on measure preserving transformations like weak mixing,
multiparameter semigroups, vector valued ergodic theorems, and the ergodic
theorem for information are postponed although they could be read right away.
§1.1 Von Neumann's mean ergodic theorem, ergodicity
1. Definitions and examples. A measurable space (Ω, si) consists of a non empty
set Ω and a σ-algebra «β/, i.e., a non empty class of subsets of Ω, closed under the
formation of complements and countable unions. A measure μ on (Ω, si) is a non
negative set function μ: si -* R+ и {oo} with μ(0) = 0 which is σ-additive. The
triple (Ω, si, μ) is called a measure space. In the case μ(Ω) = 1, μ is called a
probability measure, and (Ω, si, μ) a probability space.
Let (Ω, si) and (Ω', si') be measurable spaces. A mapping τ: Ω -* Ω' is called
measurable (or si — j/'-measurable) if τ'1 si' = {τ~ι A': A'e si'} a si. τ is
called a homomorphism of (Ω, si, μ) into (Ω'? si', μ') if τ is measurable and the
measure μ ο τ'1 defined on si' by (/in-1) (A') = μ(τ_1 A') agrees with μ'.
τ: Ω -> Ω is called measure preserving if it is measurable and satisfies /in-1
= μ. In this case μ is called invariant or τ-invariant. A measure preserving
transformation will also be called endomorphism (of (Ω, si, μ)). If τ is an invertible
endomorphism of Ω onto Ω for which τ-1 is an endomorphism, then τ is called an
automorphism.
Examples of endomorphisms of measure spaces turn up in many branches of
mathematics. Perhaps the simplest examples are translations in Rn and rotations
χ -* χ + a (mod 1) in [0,1 [ with Lebesgue measure and, more generally,
translations in locally compact groups Ω with left Haar measure.
Another class of measure preserving transformations in the same space is given
by the continuous grouptheoretic automorphisms of Ω.

2 Measure preserving and null preserving point mappings
A simple number theoretic example can be defined via the expansion
of ω β Ω = [0,1] \ Q as a continued fraction: Identify ω with (ωΐ9 ω2?...) e Ww,
where
ω= ι—/ + /—' + /—I +....
/ωχ /ω2 /ω3
Now ω -* τω = (ω2? ω3,...) defines an endomorphism in (Ω, .я/, μ) when μ is
the measure with density (1 + x)'1 with respect to Lebesgue measure; see e.g.
Billingsley [1965].
If Ω is a compact Hausdorff space and τ: Ω -* Ω continuous, there always
exists an invariant measure on the σ-algebra sf of Baire sets. As we want to
return to this example later and a proof is simple if we make use of Banach limits,
we take this liberty.
A Banach limit L is a linear functional defined on the space С of bounded
sequences χ = (χθ9 xi9...) of real numbers such that
(i) L{x) ^ 0 holds for all χ with xt ^ 0 (i = 0,1,...),
(ii) L((x1?x2,x3,...)) = L((x0,x1,x2,...)) (*e4) and
(iii) L((l,l,l,...)) = l.
Banach limits exist; see theorem 3.4.1. Using a fixed Banach limit L and a fixed
ω β Ω, we can define a positive linear functional μω on the space C(Q) of
continuous functions on Ω by μω(/) = £((/(τηω))£°=0).
By the Riesz representation theorem (see Bauer [1981]) this linear functional is
of the form μω (/) = \f(r\) μω (άη) for some measure μω on s/. The properties of L
imply that μω is an invariant probability measure.
The classical examples of endomorphisms which have originally motivated the
search for ergodic theorems arise in statistical mechanics. A theorem of Liouville
asserts the invariance of the 6r-dimensional Lebesgue measure under the Ha-
miltonian flow in phase space, see Khintchine [1949].
Still other examples can be found in § 1.4. A rich collection of examples is given
in the book of Cornfeld, Fomin, Sinai [1982].
If (Ω, s/9 μ) is a measure space, S£p = $£v (Ω, sf9 μ) denotes the space of real
or complex valued measurable functions / with \\/\\ρ'·=($\/\ράμ)1,ρ<oo,
(1 ^ ρ < oo). <£^ denotes the space of measurable functions for which
||/||oo:=inf{a>0:iu({|/|>a}) = 0} is finite. We shall also write &ρ(μ) or
Ур{£#) if we want to mention the underlying measure or σ-algebra. {\f\ > a} is a
shorthand for {ω e Ω: |/(ω)| > α}. We shall use such a shorthand notation also
for other sets defined by properties of functions. Frequently we abreviate even
further and write μ(|/| > ос) for μ({\/\ > ос}).
Recall that/= g (mod μ) means //(/=t=^) = 0, and that equality mod μ is an
equivalence relation in the space of measurable functions and in each space jSfp.
The space Lp = Lp{Q, s/9 μ) of equivalence classes in jSf p is a Banach space with
norm || · |L.

Von Neumann's mean ergodic theorem, ergodicity 3
Most of the time we shall not distinguish between elements fe Lp and their
representatives. In all statements involving only a sequence of elements of !£v or
Lp and holding only mod μ, the difference is irrelevant.
The function \A which is 1 on Α α Ω and 0 on the complement Ac is called
indicator function of A. A and В are equal mod μ if the measure of their
symmetric difference AAB = (An Bc) и (Ac η Β) is zero. Again, sets which are equal
mod μ will usually not be distinguished. 1 denotes the function = 1.
A measure ν on s/ is called μ-continuous if ν (A) = 0 holds for г\\ А e si with
μ (A) = 0. We then write ν <ξ μ. Call ν equivalent to μ (ν ~ μ) if ν <ξ μ <ξ ν.
If (Ω, si) is a measurable space and τ: Ω -* Ω a measurable mapping, the
operator/-» Tf ··=/<> τ is called composition with τ. It is a linear operator in the
space of measurable functions on Ω.
A measurable map τ: Ω -* Ω with μ ο τ"1 <ξ μ is called null preserving. An inver-
tible null preserving map τ of Ω onto Ω for which also τ-1 is null preserving is
called nonsingular. For null preserving τ the composition operator is well defined
in the spaces of equivalence classes because then/^ =/2 mod μ implies /i ο τ
= /2 °τ mod μ.
We shall make frequent use of the following notions from functional analysis:
Definition 1.1. If £, 9) are normed vector spaces and Га linear operator mapping
X into ?), the лягга || ГЦ of Г is given by
imi= sup иг/ц.
Il/ll^i
Γ is called bounded if || ГЦ is finite, and Г is called a contraction if || ГЦ ^ 1. Г is
called an isometry if || Tf\\ = ||/|| holds for all/.
If a partial order is defined in ■£, the positive cone {fe X:f^ 0} of X is denoted
by X+. Τ is called positive if Г£+ с ?)+. In the case 3; = ?) we speak of a linear
operator (contraction, ...) in X. У? (X) denotes the set of bounded linear
operators in X. I denotes the identity operator.
If τ is an endomorphism of (Ω, «я/, μ) the composition with τ is a positive isometry
in each Lp. If τ is only null preserving or nonsingular, it still is a positive
contraction in L^, but it need not map Lp into Lp for 1 ^ /? < oo.
2. Contractions in Hubert space. We denote the scalar product of two elements/,
h of a Hubert space 9) by </, A>. The dual Γ* of a bounded linear operator Tin 9)
is characterized by <7/, h} = </, Γ*Α>, valid for all/and A.
Lemma 1.2. IfT\9)-+9)isa contraction in a real or complex Hilbert space and
ge §, /Аел g—Tg holds if and only if g — T*g.
Proof If, for some g, <g,Tg} = \\g\\2, then <g, Tg> is real and <g, Tg>
= <?&£>· We then get

4 Measure preserving and null preserving point mappings
\\Tg-g\\2 = <Tg-g,Tg-g} = \\Tg\\2 + \\g\\2-2<g,Tg}
^2||g||2-2||g||2 = 0.
Thus, g = Tg is equivalent to \\g\\2 = <g, Tg} = <r*g,g>. Applying this
equivalence to Γ* the identity g = T*g follows, α
Lemma 1.3. Let $~ be a family of contractions Г ш α Hilbert space ξ). Then the
orthogonal complement F1 of F= {ge ξ>: Tg = gVTe 2Γ} is the closure of the
subspace N spanned by {h — Th: he ξ}, Те 3~}.
Proof Write g±ir)0 if g is orthogonal to a subspace §0. Now g±N о
<g, (T- I)h} = OVA egje^o <r*g -g,h} = OVA, To T*g = g\/T о
Tg = gVTogeF.
Thus TV, and hence also its closure cl TV, is orthogonal to the closed subspace F.
As any vector orthogonal to cl N belongs to F we have F1 = cl Ν. α
The following notation will be used frequently for linear operators T:
SJ-= S„(T)f-= "Σ T'f, AJ-= Ая(Т)/'=п-1 S„(T)f.
i = 0
If the operator under consideration is S we may also write A J'for An{S)f If Г is
the composition with τ we sometimes write Αη(τ) for An(T).
Theorem 1.4 (Mean ergodic theorem of von Neumann). If Τ is a contraction in a
Hilbert space ξ>, and Ρ the projection onF — {g e 9)\ Tg = g), then A Jconverges
in norm to Pfforfe §, (n -> oo).
Proof If/= (T- I)h for some A then
\\AJ\\ = л"11| ΓΑ - A + Γ2A - Th + ... + Tnh - Τη~χA||
= «-1||ΓηΑ-Α||^2«-1||Α||^0.
By approximation this yields |M„/|| -* 0 for all/in the closure of (Γ— /)§, and
now the assertion follows from Lemma 1.3. α
If τ is an endomorphism in (Ω, si, μ) and/? = 2, it follows that for any/e Lp there
is an/e Lp with/=/o τ and ||/||ρ ^ ||/||p such that ΜΒ(τ)/-/||ρ -* 0 (η - oo).
Approximation arguments (or theorem 2.1.1) show that this remains true for
1 <p < oo. If μ(Ω) = со the analogous statement does not always hold for ρ = 1;
e.g., take Ω = R1, μ — Lebesgue measure, τω = ω + 1, and /= 1[0,ΐ[· For
fe L^, \\An(T)f—/||да -* 0 need not even hold in the case μ(Ω) < со. One can
take Ω = [0,1 [, μ = Lebesgue measure, τω = ω + α (mod 1) with an irrational
a, and a suitable (highly discontinuous)/e Loo· We leave this as an exercise to the
reader.

Von Neumann's mean ergodic theorem, ergodicity 5
3. Absorbing and invariant sets. In many cases the property/ = /° τ of the limit
function can be used to find the explicit form of/ For future use we discuss the
relevant notions in some more generality than needed here.
Definition 1.5. Let τ be null preserving in (Ω, sJ, μ). A set A e si is called τ-
absorbing if Α α τ'1 A, τ-absorbing mod μ if μ(Α\τ~ί A) — 0, τ-invariant if
τ~χΑ = A, and τ-invariant mod μι{μ(ΑΑτ~ίΑ) — 0. An j^-measurable function
/is called τ-invariant if/ = /° τ and τ-invariant mod μ if μ(/Φ/° τ) = 0.
Thus A is τ-absorbing if no orbit ω, τω, τ2ω,... starting in A leaves A. If A is τ-
00
absorbing mod μ the set yl1 = yl \ (J τ"*(/1 \τ_1 yl) differs from A only by a set
fc = 0
of measure 0 and is τ-absorbing. Similarly, if A is τ-invariant mod μ the set A2
00
= (J τ~ΗΑ1 is τ-invariant and equal to yl mod μ. Thus, in all considerations
fc = 0
where sets of measure 0 do not matter, we need not distinguish the notions τ-
invariant and τ-invariant mod μ. A real valued / is τ-invariant if and only if
{/> a} is τ-invariant for all α e R. A complex valued/is τ-invariant if both the
real and the complex part are τ-invariant. These observations can be used to
show that for functions/which are τ-invariant mod μ there exists a τ-invariant/'
with μ(/Φ/') = 0, and we need not distinguish the notions τ-invariant and τ-
invariant mod μ for functions either.
For any null preserving τ the family J of invariant sets clearly is a σ-algebra.
By the above remarks a function/is τ-invariant if and only if it is «/-measurable.
If τ is an endomorphism and μ finite, the σ-algebra J can be used to to express the
limit/of yln(Y)/as a conditional expectation'.
Let ^ be a sub-a-algebra of sJ, such that the restriction of μ to 3F is σ-finite.
Recall that for any /e Lx (Ω, sJ, μ) there exists (by the Radon-Nikodym
theorem) an/0 e Lx which is ^-measurable and satisfies
Ιίοάμ^Ι/άμ V,4eiF,
A A
and that/0 is uniquely determined mod μ./0 is called the conditional expectation
of/with respect to J^, and denoted by E(f\ J^) or by Εμ(/\ ^) if we emphasize
the fact that the basic measure is μ. It follows from the Jensen inequality that the
map/-> E(f\tF) is a contraction E# in each space ΕΡ(Ω, sJ, μ).
Proposition 1.6. If τ is an endomorphism in (Ω, si, μ), and μ σ-finite on the σ-
algebra J of τ-invariant sets, then, for any fe L2(0, si, μ), the norm-limit J of
AnWfisgivenbyf=E(f\J).
Proof. We may assume μ(Ω) < oo. /is «/-measurable. For any A e J we have

6 Measure preserving and null preserving point mappings
J /ο τκάμ = J /ϋμ because yl is τ-invariant and τ an endomorphism. Now J /φ
A A A
= (An(x)flA} -► <^1^> = J/Φ because strong convergence implies weak
convergence, α Α
Definition 1.7. A null preserving transformation τ in (Ω, si,μ) is called ergodic if
all τ-invariant sets yl have the property that μ (A) = 0 or μ (A*) = 0.
Thus, τ is ergodic if the space cannot be decomposed into two non trivial τ-
invariant subsets. If an endomorphism is ergodic and 0 < μ(Ω) < oo,the limit/is
simply given by/= μ(Ω)-1 \/άμ. This means that for large η the space average
μ(Ω)"1 J/ί/μ is very close to the time average An{x)f The important ergodic
hypothesis in statistical mechanics is the assumption that these two averages are
asymptotically equal for the endomorphisms arising in the Hamiltonian flow in
phase space. Von Neumann's theorem made it clear that the limit of the time
averages does exist in the sense of L2-convergence and that it is equal to the space
average in the ergodic case. The question of ergodicity of the transformations
constituting the Hamiltonian flow was left open. For many endomorphisms the
proof of their ergodicity is fairly simple, but for some, including the
endomorphisms coming from the Hamiltonian flow, the question whether they are ergodic is
very deep; see the Notes.
4. Criteria of ergodicity. We end this section by describing some general
necessary and sufficient conditions for ergodicity. They do not go very far beyond
reformulations of the definition. The proof of the ergodicity for specific examples
usually requires arguments which exploit the specific nature of the examples.
(a) For general null preserving τ we can just say that τ is ergodic iff each
measurable τ-invariant function/is constant μ-a.e. This follows from our above
remarks on τ-invariant sets and functions. (As usual "iff" means "if and only if).
(b) If τ is nonsingular, τ is ergodic iff μ (A) > 0 implies that the complement of
+ 00
A* = (J T~kA'\sdL nullset. A* is the smallest τ-invariant set containing A. An
k= - oo
obvious equivalent condition is that μ (A) > 0, μ (Β) > 0 imply the existence of an
integer к with μ{τ*ΑηΒ)> Ο.
Proposition 1.8. For an endomorphism τ of a finite measure space (Ω, si, μ) each
of the following conditions is equivalent to the ergodicity of v.
(el) μ(Α)>0 => μ((Λ")<) = 0, where А~ = Q x~kA;
k = 0
(e2) μ(Α)>0, μ(Β)>0 => 3k ^ 0 with μ(τ~1ίΑηΒ) > 0;
η-ί
(e3) For all A, Be si, limn'1 Σ μ(τ~1ίΑηΒ) = μ(Α)μ(Β)μ(Ω)-1;
n->oo fc = 0

Birkhoff's ergodic theorem 7
(e 4) There exists a family Μ с: si such that the linear combinations of the
functions \E {Ее Л) lie dense in L2 and which has the property that Αη{τ)\Έ
converges weakly in L2 to a constant;
(e5) For allf,geL2, lim и"1 "ς <f°*,g> = </J> <1^>μ(Ω)-ί.
n->oo fc = 0
Proof The equivalence of (el) and (e2) to ergodicity follows because
τ~χΑ~α:Α~ and μ(τ~χ Α~) = μ(Α~) imply μ(Α~\τ~ίΑ~) = 0. Von
Neumann's theorem yields the existence of the limit in (e 5) (and hence in (e 3) and
(e 4)) and the limit is </, g}. If τ is ergodic/is constant so that the identity (f 1 >
= </, 1> implies/= μ(Ω)~1 </, 1>. But <fg> = </, 1> <bg> μ(Ω)"1. Taking
/= 1л? g = 1B we get (e 3). If τ is not ergodic, (e 3) and hence (e 5) cannot hold
because (e 2) doesn't. The equivalence of (e 4) is obtained by an approximation
argument. D
Notes
Koopman [1931] observed that, using Liouville's theorem and the measure preserving
property, the Hamiltonian flow could be studied via the induced group of unitary
operators in Hubert space. This idea led von Neumann [1931] to a proof of his ergodic
theorem via spectral theory.
The measure theoretic concept of ergodicity seems to go back to Birkhoff and Smith
[1928], who used the term metrically transitive, still favoured by some authors.
Sinai [1963] announced a theorem which - roughly speaking - says that a system of η
balls of equal diameter following the laws of elastic reflection in a cubic box is ergodic. No
published proof of this result seems to exist. Sinai [1970] considered the movement of only
one ball in a 2-dimensional domain with smooth strictly positive curvature (dispersing
billiards). This was simplified and generalized by Bunimovich-Sinai [1973]. Also Kubo
[1976], Kubo-Murata [1981], Gallavotti [1975], and Keller [1977] contributed to this
subject.
Assuming, in addition, "finite horizon", Gallavotti and Ornstein [1974] proved that
dispersing billiards are isomorphic to Bernoulli shifts.
We refer to Bunimovich [1982] for a survey on recent developments in this area.
§1.2 Birkhoff s ergodic theorem
1. Discrete time. Our next aim is to give a proof of George D. BirkhofFs famous
pointwise ergodic theorem. For later use we formulate some of the arguments in
the more general operator theoretic setting. We write
(2.1) Mns/=Max(51/?...5iSJ), MJ=Mvl{AJ9...9AJ),
and
Λ4/= sup^M,/.

8 Measure preserving and null preserving point mappings
When this notation is applied to an endomophism τ, Τ will be the composition
with τ.
The key step in the proof of the pointwise ergodic theorem is the following
maximal ergodic theorem, which was discovered for endomorphisms by Yosida-
Kakutani [1939] and for general positive contractions in Lx by Hopf [1954]:
Theorem 2.1. Let Τ be a positive contraction in Lx (Ω, si, μ). For real valued
fe Lx put En = {Mnf^ 0} (= {Msnf^ 0}) and ^ = Q En. Then
n=l
l/άμ^Ο and J/ί/μ^Ο.
Ьп Еоо
Proof. We follow the elegant argument of A.M. Garsia [1965]: For
к = 1,..., и, (Msnf)+ ^ SJtmd hence/+ T(Msnf)+ >f+ TSJ= Sk+1f. Thus
/^ SJ- Τ(Μξ/)+ holds for к = 1,..., η because it is trivial for к = 1. Passing
to maxima one obtains/^ Mlf— T(M%f)+. Now integrate over En\
En En
= |((м^)+-г(м^л+)Ф
En
= $(Μξβ+άμ-\Τ(Μξβ + άμ
Ω Εη
Ω Ω
because ^άμ ^ J Τΐηάμ holds for h e L\. A passage to the limit yields the second
inequality, α
This proof is a bit miraculous. In section 3.2 a discussion of the filling scheme will
provide a longer but more intuitive proof.
The following maximal ergodic inequality was already known to Wiener
[1939]. It admits also a simple direct proof (see theorem 5.2), and suffices for the
proof of BirkhofTs theorem given in theorem 7.3.
Corollary 2.2. If τ is an endomorphism in a measure space (Ω, si, μ), then
holds for any real valued fe Lx and a > 0.
Proof It is enough to prove \\f\\i ^ αμ(Α) for arbitrary sets A of finite measure
contained in {MJ^ a}. Put EnA = {Mn(f- alA) ^ 0}. By theorem 2.1,

Birkhoff's ergodic theorem 9
j (/-а1л)ф^0.
En,A
For any сое Acz {Mnf^ a} there exists ak ^n with Akf{cj) ^ a. This implies
Sfc(/- αϊ) (ω) ^ 0 and 5fc(/- αϊΑ) (ω) ^ 0. Thus A is contained in £Bti4, and
ll/lli^ Ι ίάμ>* f ίΛάμ = αμ(Α). Π
En,A En, A
Theorem 2.3 (BirkhofFs ergodic theorem). 7/4 w ал endomorphism in a measure
space (Ω, «я/, μ) andfe Lx (real or complex), then the averages Anf converge μ
— a.e. to some τ-invariant f with ||/||i^||/Hi. For each τ-invariant Ae si with
μ (A) < oo
(2.1) f/rf|i = f/rf|i.
A A
Proof. First consider a real valued /. Because of
AH+J=(n + i)-iSH+J=(n + iyif+-^-T(-Sj)oT
η +1 \n J
the functions/" = lim supn_^00yln/and// = lim inf^^^/are τ-invariant. We
show that/" and/* assume the values ±oo only on a set of measure 0:
For any β > 0 the τ-invariant set Όβ = {/" > β} is contained in the union of
the increasing sequence {Mnf^ β}. By the maximal inequality μ(Μη/^ β)
й β~χ ML· Hence μ(Όβ) й β'1 ll/lli- Passing with β to infinity/" < oo a.e.
follows. By symmetry fl > — oo a.e., and μ(/1 < α) = μ{— lim sup An(—f) < a)
^lam/Hi for a<0.
If ylw/does not converge a.e. there exist rational numbers α < β such that the
set В = {fl < α < β </"} has positive measure, μ (Β) cannot be infinite because
а < 0 or β > 0. By the τ-invariance of В the function f = (/— β)1Β has the
property that /' ° тк vanishes outside В for all к ^ 0 and Β = {ω: Зп
^1 mthSJ'(a>)>0}.
The maximal ergodic theorem implies βμ(Β) <; \/άμ.
в
A symmetric argument with/" = (a —/) 1B gives us \fdμ S αμ(Β). Together
в
this contradicts α < β.
We have shown that/" =/' a.e., and that these functions μ — a.e. assume only
finite values. By the decomposition/ = /4" —f~ we may assume/e L\ for the
proof of 11/11 x ^ 11/111# The lemma of Fatou then yields
\/άμ = \\\m'mf Αη/άμ rg liminf \Α^άμ = \/άμ
where the last equality follows from J/° τ4άμ = \/άμ.

10 Measure preserving and null preserving point mappings
To prove the last statement in the theorem we may assume Ω = Α, μ{Ω) < oo,
and/^ 0. Foranye > 0 there exists a Κε ^ 0 such that gg =/— (/л Κε) has norm
HgJI^e. Now
\(Anf- Κε) + άμ й $Α^εάμ < ε
shows that the sequence ^4„/is uniformly integrable. As any uniformy.integrable
sequence converging μ-a.e. converges in I^-norm, the assertion \/άμ = \/άμ
follows from §Anf= \/άμ.
For complex valued / one can use the decomposition into the real and the
imaginary parts to prove the existence of the limit, and observe \f\
uKmAn\f\. D
The condition (2.1) determines/uniquely: We can assume that μ is σ-finite. There
exists a set Q0e/ such that every AeJ with μ (A) < oo is contained mod μ in
Ω0, and μ is σ-finite on Ω0 η «/./vanishes in Ω°0 and is given by E{f\J>) in Ω0. In
particular, if τ is an ergodic endomorphism in a σ-finite infinite measure space,
then/must vanish for all/.
2. Continuous time. We have stated the ergodic theorem for a single
endomorphism τ. Sometimes one is interested in a continuous time motion of the points ω,
and in a corresponding continuous time theorem. There is no difficulty to derive
such a result from theorem 2.3. By a/W {τί? /eR}we mean a group of
measurable transformations xt: Ω -* Ω with τ0 = identity, xt+s = xt ° τ5, (/, jeR). The
flow will be called measure preserving if the xt are measure preserving. The flow is
called measurable, if the map (ω, t) -* τ, ω from fixR1 into Ω is JZ — si-
measurable, where si is the completion of the products-algebra si ® 36 of si
with the Borel sets, and the completion is taken with respect to the product μ of μ
on si and the Lebesgue measure λ on 36, The completely analoguous definitions
can be given for semiflows {τί? t ^ 0}, (in which the xt need not be invertible). If
{τί? t ^ 0} is a measurable measure preserving semiflow in a σ-finite measure
space, and if/ Ω -* R is integrable, the function/defined by/(ω, /) —f{ztd) is,
for all T> 0, integrable in Ω χ [0, T~] (Fubini), and, hence, for μ-a.e. ω e Ω the
Τ η n-l
integrals J f(xtoS)dt are well-defined. Note that \ f(xtoS)dt = Σ Ρ(τιω) with
ο ο ί=ο
ι
F(co) = \f{xt(D)dt and that Fis integrable. Therefore the ergodic theorem im-
o
η
plies that n~x j" f(ztd)dt converges a.e. when the integers η tend to infinity. The
ergodic theorem also implies n~1F0 ° τ""1 -* 0 a.e., where F0 = J \f°zt\dt. For η
Τη °
^ T<n + 1, | \f°Ttdt— \f°ztdt\ rg Fqot"-1. Thus the convergence a.e. of
т о о
Γ-1 J /° τ, Λ when Г tends to infinity along the reals is a rather trivial conse-
o

Birkhoff's ergodic theorem 11
quence of the discrete time theorem. Many of the discrete parameter theorems in
this book will have such continuous parameter analogues and we shall usually
not care to state the latter.
There is, however, another class of continuous parameter theorems which will
be of more interest to us. These are the local ergodic theorems. They assert the
convergence of continuous time averges over time intervals [0, ε] when ε -* 0
+ 0. They were introduced by N. Wiener [1939] for measurable measure
preserving flows. In this case they are a consequence of a form of the fundamental
theorem of calculus, which says that for Lebesgue integrable/on [0, oo [ one has
lim-'j7(')<fc=/(')
ε->0 β s
for Λ-almost all s; see Royden [1968: Ch. 5].
Theorem 2.4 (Wiener's local ergodic theorem). If(Q, stf\ μ) is a σ-finite measure
space and {τί? t ^ 0} a measure preserving measurable semiflow,fe <£x (μ), then
ε
lim ε-1 J f(Taco)d<x —/{ω) holds μ-a.e..
£-►0 + 0 0
Proof. Let TV denote the complement of the set of points (ω, /) in Ω χ [0, oo[ with
ε ^
lim ε"1 |/(τί+αω)ί/α = /(т,со) and let Νω = {te [0, oo[: (ω, ή e Ν} and Ν1
ε->0 + 0 0
^ t
= {ω 6 Ω: (ω, t)e Ν}. For μ-almost all ω, |/(ταω)ί/α is well defined for all / and
о
/(ω, α) is integrable in each [0, T~]. The fundamental theorem of calculus implies
λ(Νω) = 0 for these ω.
μ(Ν) = 0 follows, and, by Fubini, Λ-almost all / have the property that μ(Ν*)
= 0. But A/* = τ"1 JV°. As xt is measure preserving μ(№) = 0 follows, α
The integrability off (ω, ·) with respect to λ is also clear for bounded/, when the
xt are not necessarily measure preserving. If μ (Ν1) — 0 for almost all / > 0 and the
τ, are null preserving μ{Νί) — 0 follows for all / > 0. Therefore the above
argument also proves:
Theorem 2.5. Ififi, $ί, μ) is a σ-finite measure space and {τί? / ^ 0} a null
preserving measurable semiflow with the property that
μ(τ^1 A) — 0 for all t>0 implies μ (A) = 0
then
ε
lim ε-1 Jf(xt(o)dt = /(ω) μ-a.e.
ε->0 + 0 Ο
holds for all bounded measurable f

12 Measure preserving and null preserving point mappings
3. Uniform convergence. Simple examples like the Bernoulli shifts (§ 1.4) show
that ^4„/need not converge everywhere even when Ω is a compact topological
space, and τ and/are continuous. However, for some interesting examples one
obtains even uniform convergence by the following special case of the mean
ergodic theorem in Banach spaces:
Theorem 2.6. Let Ω be a compact metric space with metric ρ. If τ: Ω -> Ω is
continuous, and the functions Anf (n ^ 1), are equicontinuous, then Anf converges
uniformly in Ω.
(Recall that the functions/,, are called equicontinuous if for any ε > 0 there exists
δ > 0 such that ρ (ω, ω') < δ implies |/„(ω) —/„(ω') I < β for all n).
Proof Let (5 (ω) denote the closure of the orbit {ω, τω,...} of a point ω. If some
g e 0(Ω) is 0 on 0 (ω), then μω(#) = 0, where μω is the linear functional
constructed in subsection 1 of § 1.1. Thus μω(Θ(ω)) = l._
By the Birkhoff theorem there exists an ω* e Θ (ω) for which Anf(co*)
converges to some finite/(ω*). Given ε > 0 find δ > 0 by the equicontinuity of the
Anf and then find m with ρ(τ/ηω, ω*) < δ. \Anf((o) — ^1п/(ттсо)| tends to 0 for
η -* oo, since all but 2m terms in the sums cancel and/is bounded. \Anf{TmaS)
— Anf(co*)\ < ε implies \Anf(coi) — /(ω*)| < 2ε for large n. As ε was arbitrary
lim Anf((D) exists.
By the compactness of Ω there exist ω1?..., ωκ β Ω so that the open δ — balls
Bk with these centers cover Ω. If TV is large enough, then \Anf(co^) —f(ojk)\ < ε,
for η Ξ> N and к = 1,..., Κ. If ω is arbitrary there exists A: with ρ (ω, cok) < (5. Now
\Anf(cu) — Anf(co^\ < ε shows that the convergence is uniform. D
Example. Consider the d-dimensional torus Ω = [0,1 [d, which is a compact
group with coordinatewise addition mod 1. Take some α = (α1?..., ad) e Ω, for
which α1? α2?..., ad? 1 are integrally independent, i.e., mx = m2 = ... = md = 0
d
shall be the only integers for which Σ °^ктк is-an integer.
fc = l
The translation τ: ω -> ω Η- α clearly preserves the ^-dimensional Lebesgue
measure μ = λά. τ is ergodic. This can be proved using Fourier analysis; see e.g.
Petersen [1983: p. 51]. We sketch a proof using only measure theory. First show
by induction on d that the sequence ..., — 2a, — a, 0, a, 2a, 3a,... is dense in Ω:
As all these points are different there is an accumulation point, and, hence, for
any ε > 0, а к e Ζ with 0 < | ol[ \ < s/2d, where ol[ = koct mod 1. Because of the
induction hypothesis the line {χα': xeR} through a' = (a'1?..., a^) and
(0, 0,..., 0) lies densely in Ω. (It intersects {0} χ [0,1 [d~l in a dense subset.) For
any point ζ on this line there is some ma' (m e Z) in an ε/2-neighbourhood of z.
Thus, for any у e Ω, there is some mk<x in the ε-neighbourhood of y. (A similar

Birkhoff's ergodic theorem 13
argument shows the density of the "forward orbit" a, 2a,..., which was already
known to Kronecker [1884]).
Now let A be a τ-invariant set of positive measure, and let 0 < η < 1 be given.
For 5, Ce si we say that В fills out (1 -η) of С when/^jB η С) ^ (1 - ι/)/ι (С). If
ξ > 0 is sufficiently small one can fill out (1 — η) of Ω with say к (ξ) disjoint cubes
of side-length ξ which lie at a strictly positive distance of each other. There are
arbibrarily small ξ such that A fills out (1 — η) of some cube Q of side length ξ. By
the density statement above, we now can find fc(£) disjoint translates τ* Q of Q.
As τ1 A = A, A fills out (1 — η) of τ* Q, and, hence, (1 — η)2 of Ω. As т/ > 0 was
arbitrarily small μ(^4) = 1, and the ergodicity of τ follows.
An application of theorem 2.6 to this example now proves the theorem of Weyl
[1916] on uniform distribution mod 1:
Theorem 2.7. Ifτ in Ω = [0,1 [d is the translation mod ltya = (a1,...,a(i) and
the numbers a1?..., ad, 1 are integrally independent, then, for each continuous f
Anf converges uniformly to ]Υ#μ.
Proof It is simple to check that the yl„/are equicontinuous. By the ergodicity the
limit must be the constant \fdμ. α
Sometimes Weyl's theorem is spelled out for Riemann integrable functions, but
this is really an equivalent formulation which can be obtained by a simple
approximation. (To prove convergence a.e. for/e JSfx one needs a maximal inequality
even in this special case).
Notes
Birkhoff [1931] has based his proof on the following (weaker) maximal inequality: If Fa is
the τ-invariant set {Hmsupv4n/^ a}, then \ fdμ ^ αμ(7£).
He actually has formulated his a result only for indicator functions/ = 1B (in the setting
of a closed analytic manifold having a finite invariant measure). Khintchine [1933] then
showed that Birkhoffs result remained true for integrable/on an abstract finite measure
space. (Therefore theorem 2.3 is called Birkhoff-Khintchine-Theorem in a few countries.
However, Khintchine himself emphasized that the idea of his proof was precisely that of
Birkhoff).
BirkhofFs theorem (for finite μ) easily implies norm convergence of ^4„/in Lp for/e Lp
(1 ^ ρ < oo), and therefore contains the special case of von Neumann's theorem which
motivated his work. As BirkhofFs paper appeared earlier, it is of interest that von
Neumann's theorem was proved first and was known to Birkhoff.
The original proof of Yosida-Kakutani's maximal ergodic theorem, which used ideas of
Kolmogorov [1937], remains of interest; see Petersen [1983]. Simplified and alternative
arguments (sometimes for continuous time or special cases) were given by Riesz [1931],
[1932], [1942], [1945], Pitt [1942], Hopf [1947], Dowker [1950] and others. Kamae
[1982] gave a proof using nonstandard analysis. Using some of his ideas, Katznelson and
Weiss [1982] gave a short proof without explicit use of a maximal ergodic theorem.

14 Measure preserving and null preserving point mappings
E. Bishop [1966], [1967], [1968] gave a proof of BirkhofFs theorem using upcrossing
inequalities similar to those in martingale theory; see § 4.1. These inequalities are
constructively valid, but BirkhofFs theorem is not. Nuber [1972] has sought conditions which
constructively imply the conclusion of BirkhofFs theorem.
The first ratio ergodic theorem was proved by Stepanov [1936] (in the ergodic case) and
Hopf [1937]: If τ is an endomorphism of a σ-finite (Ω, sf, μ),/ε J£\ and g e <£\+, then
Snf/Sng converges a.e. on each {Skg>0}. This is now a special case of the Chacon-
Ornstein theorem.
It is easy to see that BirkhofFs theorem implies the convergence a.e. of
+ Л
Bxnf-.= (2« + 1)"1 Σ e2ltikXf° Tk for fixed λ e R. This was strengthened by Wiener and
k=-n
Wintner [1941] who proved:
Theorem 2.8. If τ is an automorphism of a finite measure space, then there exists, for each
fe&ua nullset Nf such that, for ω e TV}, Bxf(co) converges for all λ e U.
Wiener and Wintner [1941 a] also investigated when F(t) = /(τ, ω) is almost periodic and
4-1 _ _
studied lim (It)'1 J F(s + u)F(s)ds, where Fis the complex conjugate function.
-ί
Speed of convergence. When a convergence statement has been proved, one of the
questions of interest is whether one can assert something about the speed of convergence. The
famous law of the iterated logarithm is an example where a positive result is possible. In
the form proved by Hartman and Wintner [1941] it can be stated in our language as
follows: If μ(Ω) = 1, τ is an endomorphism, Tf = f° τ,/has integral \ίάμ = 0 and L2-
norm ||/||2 = 1, and if/,/n,/°τ2,... are independent, then
lim sup Anfl]/l\og\ognjn = 1 μ-a.e..
By symmetry liminf = —1, so that Anf = О ((и-1 log log и)1/2) a.e. Independence is
crucial! No general positive ergodic theoretic result of this type is possible even for slower
speeds. Indeed, Krengel [1978a] has shown: If τ is an ergodic endomorphism of the torus
[0,1 [ with Lebesgue measure, and (a„) any null sequence of positive numbers, there exists
a continuous f with integral 0 and
(2.2) lim sup a"1 \Anf\ = 00 a.e..
On the other hand, Halasz [1976] proved: For any non decreasing sequence (cn) of
positive numbers with c1 ^ 2 and tending to 00, and for any ergodic automorphism of [0,1 [
there exists A with λ(Α) = \ and | Sn 1A — n/2\ ^ cn for all n. Thus, the convergence can be
arbitrarily fast.
The following deep theorem of O'Brien [1983] contains a limit version of Halasz' result
and (2.2) for measurable/:
Theorem 2.9. If (bn) is a sequence of positive numbers tending to 00 and satisfying
lim inf «_1 bn = 0, there exists τ and a { +1, — \}-valuedf with lim sup b'1 Snf= 1, and this
f can be constructed in such a way that the sequences f /n,/n2, ... and —/ —/° τ,
—/° τ2,... have the same joint distribution.
Kakutani and Petersen [1981] proved another strengthening of (2.2) for measurable/
They construct, for any sequence (bn) of positive numbers with divergent sum, a bounded
к
measurable / with integral 0 and sup| Σ b{AJ\ = 00 a.e. Dowker and Erdos [1959]
к ί=1

Birkhoff's ergodic theorem 15
к
showed the existence of a bounded/with \fdX — 0 for which Σ bj° τ1 fails to converge in
measure on any subset of [0,1 [ having positive measure, and with sup | X btf° τ* | = oo
a.e., see also Halmos [1948]. Baum and Katz [1965] showed: Σμ(\Αη/\ > ε)/η converges
for /with \/άμ = 0 if/,/° τ... are independent.
The uniform boundedness principle implies that the existence of a sequence an tending
to oo, with lim sup aB || Anf— Pf \\ < oo for all/, is equivalent to || An — P\\ -» 0. It is
therefore easy to see that there is no speed of convergence in von Neumann's mean ergodic
theorem.
There are a few positive results on speed of convergence for specific transformations
and functions; see e.g. Kuipers and Niederreiter [1974], or Kowada [1973].
Non-integrable functions. If τ is an ergodic endomorphism in a probability space, and a
non negative/has a infinite integral, then BirkhofFs theorem implies lim Anf= oo,
because n^°°
liminfv4„/^ lim Ап(/л к) = J (/л k)d\i for all k.
n~* oo n-*ao
J. Aaronson [1977] has proved that also other norming factors of Snf than n'1 cannot
produce convergence a.e. to a non zero finite function:
Theorem 2.10. If τ is an ergodic endomorphism of a probability space (Ω, л/, μ),/ ^ 0, and
\ίάμ = oo, then for any sequence (bn) of positive reals, one has either
(2.3) lim sup b'1 Snf = oo a.e. or
л-»оо
(2.4) liminf^15'n/=0 a.e..
n-»oo
There is a similar theorem for σ-finite measure spaces; see also Aaronson [1979].
Aaronson [1981] also considered this problem for functions/which need not be non
negative: If bjn is non decreasing and tends to oo, then (2.3) or (2.4) hold with Snf
replaced by \Snf\. In the case where the/° τ* are independent Feller [1946] proved this
assertion for arbitrary non decreasing sequences. But in the general ergodic case the
condition bjn -» oo cannot be deleted: Aaronson [1977] gave an example of an/with
J l/l άμ = oo and Snf/n -»1 a.e. A related result is that of Kesten [1975], who showed that
lim inf Anf> 0 a.e. if Snf-+ oo a.e..
Let (an) be a non decreasing sequence of positive numbers. Tanny [1974] showed that
the condition lim inf ak. Jan > 1 for some k > 1 implies that for any ergodic τ in a
probability space and all/, lim sup/° τη/αη — oo a.e. or lim sup/° τη/αη = 0 a.e. O'Brien [1982]
proved that the condition is also necessary.
Dowker and Erdos [1959] have given several more examples: E.g. they showed that
SnWg/Snit^g таУ diverge for ergodic automorphisms of a σ-finite measure space.
Del Junco and Steele [1977] have shown that for any ergodic endomorphism τ in [0,1]
(which Lebesgue measure) and for any increasing sequence 0 < bx ^ b2 й of integers bn
n-fb„
with n~^bn -» 0 there exists an indicator function/= \A such that lim sup b'1 Σ /° τ1
i = n
= 1 μ-a.e. and the corresponding lim inf is 0 μ-a.e. (Wiener and Wintner [1941a] have
made a similar observation; see also PfarTelhuber [1975]).
By an example of Burkholder [1962] the averages considered in BirkhofFs theorem

16 Measure preserving and null preserving point mappings
may diverge a.e. after an application of a single conditional expectation operator. Isaac
[1973] has studied similar questions for ratios.
We refer to Kuipers and Niederreiter [1974] for the theory of uniform distribution
modi.
For abstract ergodic theorems in a Boolean algebra and in a logic, see Bunjakov [1973],
Dvurecenskij and Riecan [1980], and Pulmannova [1982].
§ 1.3 Recurrence
1. The conservative and dissipative part. We call a null preserving τ recurrent if,
00
for all Ae si\ μ-almost all ω e A belong to the set Aret = An (J x~k A of points
fc = l
returning at least once, and infinitely recurrent if, for all Ae si, μ-almost all
ω g A belong to the set
00 00
Ainf = {(oe Α: τ*ωe A for infinitely many k^l} = An f] (J x~kA.
и=1 к = п
A set We si is called wandering if the sets x~k W{k ^ 0) are disjoint, or, equiva-
lently, if no point in W returns to W. τ is called conservative if there exists no
wandering set of positive measure. Finally, τ is called incompressible, if there
exists ъо Ae si with Α α τ'1 A and μ(τ~1Α\Α) > 0. Passing to complements
one sees that τ is incompressible if and only ίΐτ~χΒ с В implies μ (ΒΧτ-1 Β) = 0.
Theorem 3.1 (Recurrence theorem). Let τ be null preserving in a measure space
(Ω, si, μ). The following conditions are equivalent:
(i) τ is conservative,
(ii) τ is recurrent,
(Hi) τ is infinitely recurrent,
(iv) τ is incompressible.
Proof, (i) => (ii): For any A the set A0 — A\Aret of points in A which never return
is wandering, and, hence, a nullset.
(ii) => (iii): For any Ae si, μ(Α0) = 0. The set т~кА0 is the set of points which
visit A at time к for the last time. If ω e A does not return to A infinitely often
00
there must be some к 2> 0 with ω e т~кА0. Hence Ainf = A\ (J т~кА0 differs
from A by a nullset. k=0
(iii) => (ii) is trivial.
(ii) => (iv): If τ-1 Β α Β, then the sequence τ ~k В, к = 1, 2,... is decreasing, and
00
Β\τ~χΒ = B\ (J x~kB is the set of points in В which never return. As τ is
fc = l
recurrent μ(Β\τ~1 Β) = 0.

Recurrence 17
(iv) => (i): If W is wandering put В = (J x~k W. Clearly τ-1 В с 5. By the
fc = 0
disjointness of the sets т"к f^we have f^ = Β\τ~χ Β. Hence μ(\ν) = 0. α
Clearly, if τ is an endomophism of a finite measure space, there cannot exist a
wandering set of positive measure, and, therefore, τ must be infinitely recurrent.
This is the recurrence theorem of Poincare [1899], perhaps the oldest result in
ergodic theory. Actually, Poincare was most interested in a topological type of
recurrence, which can be deduced from the result stated above: If Ω is a
topological space with a countable basis Μ с sf, and τ is infinitely recurrent, then almost
every соей returns infinitely often to any neighborhood of itself. To see this,
observe that the union Ε of all sets A\Ainf with A e 38 has measure 0. If U is a
neighborhood of some ω e E\ there is an A e Μ with ω e A a U. As ω φ A\Ainf,
ω returns infinitely often to A, and, hence, to U.
In general, if μ is σ-finite one can find a maximal subset of Ω on which τ is
conservative.
Theorem 3.2 (Hopf decomposition). If τ is null preserving in the σ-finite measure
space (Ω, si, μ), there exists a decomposition of Ω into two disjoint measurable sets
С and D, the conservative and the dissipative part, such that
(i) С is τ-absorbing,
(ii) the restriction of τ to С is conservative, and
(Hi) D = Ω\0 is an at most countable union of wandering sets.
If τ is even nonsingular, С is τ-invariant and there exists a wandering W0 with D
+ 00
= (J тк W0. (τ is called dissipative if Ω = D.)
k= — oo
The proof is based on an exhaustion argument. As this type of argument is used
frequently, let us explain this simple technique: Let Ρ be a certain property of
measurable sets in the σ-finite measure space (Ω, sf, μ) which is such that any
subset of a set with this property again has property P. As μ is σ-finite, there exists
a finite measure ν equivalent to μ. Put at = sup {v(A): A has property P}. Pick
some Ax e s/ with property Ρ and v(A1)7z2~1a1. lfAl9 ...9An have been deter-
n
mined, put an+1 = sup {v(A): A has property Ρ and А с Ω\ (J Ak}. Then find
fc = l
η oo
An+1 czQ\ (J ylkwithpropertyPsuchthatv(yln+1) ^ 2~1(χη+1.1,είΩ1 = (J A{.
k=l i=l
The complement Ω2 of Ωχ is such that no subset of Ω2 with positive measure has
property P. Moreover, if Ω is the disjoint union of Ω[ and Ω2 such that Ω[ is a
countable union of sets with property P, and no subset of Ω2 has positive measure
and property P, then Ωχ coincides up to nullsets with Ω[.
Proof of theorem 3.2. Take the property of being wandering and put Ωχ = Д Ω2

18 Measure preserving and null preserving point mappings
= С As τ-1 (f is wandering if W is wandering we see that τ"1 D a D up to
nullsets. Hence С is τ-absorbing mod μ. Changing С on a nullset we can assume
that it is τ-absorbing.
If τ is nonsingular, also τ W is wandering if H^is, and we can infer that D and С
are τ-invariant. W0 is constructed as follows: Take the At from the construction of
+ 00
A put Af = (J xkAb and
k= — oo
^o=Uw\U4)· °
Also for general null preserving τ the description of D can be made somewhat
more precise: Put Wx = Ax, and for η ^ 1 put
00
Wn+1 = Ан+1 и W η ( U τ-4+1)0·
i = 0
oo
Then the sequence Dn= (J τ"1 J^ is increasing with union Z>, and each Wn is
wandering. i=0
The following complement to the Hopf decomposition has been observed by J.
Feldman [1962] in a more general situation. We leave it as an exercise.
Proposition 3.3. Also in the case where τ is an endomorphism, С is τ-invariant.
Halmos [1947] proved that the powers of a conservative τ are conservative. In
fact we have:
Theorem 3.4. If τ is null preserving in a σ-finite measure space (Ω, si, μ), the
dissipative parts Dn ofxn are the same for all и ^ 1.
Proof If W is wandering for τ it is wandering for τη, so that D1 a Dn. Now
00
let Wu W2,... be wandering sets for τη with union Dn. Put h = Σ 2~kl\vk· Then
h is strictly positive on Dn and fc-1
00 00 00 00
χ Λοτ*»= Σ 2~k Σ W^ Σ 2"k^l on Ω.
i = 0 k=l i = 0 fc = l
If Dn is not contained in D1 = D, there exists an ε > 0 and A cz С with μ{Α) > 0
and h ^ εΐ^. Now
oo oo n— 1 oo
Σ Ι^τ^ε-1 Σ Α'τ^β"1 Σ (Σ Аи^н^Г'иопЙ.
ί = 0 > = 0 ./ = 0 ί = 0
But then no point in A can return to A more than ε-1 η times, a contradiction to
AcC. □

Recurrence 19
The transformation к -+ к + 1 on Ω = Ζ with the counting measure is an
example of an ergodic measure preserving invertible transformation which is
dissipative. This is essentially the only invertible example because of the last
assertion in theorem 3.2. In particular a nonsingular ergodic τ in a non atomic
measure space must be conservative. There exist endomorphisms in non atomic
σ-finite measure spaces which are dissipative and yet ergodic; see § 3.1 (Notes).
Proposition 3.5. If a null preserving transformation τ is conservative and Α τ-
absorbing mod μ, then A is τ-invariant mod μ.
Proof We may assume τ~χΑ^ A by modifying A on a nullset. Almost every
point of Ε = τ-1 A\A returns to E. But τ'1 Aid A implies that, for all ω e E, the
orbit τω, τ2 ω,... is contained in A с E\ Thus Ε must have measure 0. D
It follows that a conservative null preserving transformation is ergodic if and
only ίΐμ(Α) > 0 implies /i(( Q т~кА)с) = 0.
fc = 0
2. Induced transformations. If τ is recurrent, the return time
rA(coi)--= inf {к ^ 1:тксоеА}
to A e si is finite a.e. in A. In the measure preserving case we can evaluate the
integral of rA:
Theorem 3.6 (Recurrence theorem of Kac [1947]). If τ is a conservative endomor-
phism of (Ω, si, μ) and Ae si', then
00
\νΛ{ω)μ{άω) = μ{\] τ~ηΑ).
A n = 0
In particular, if τ is ergodic and μ (A) > 0, then § rAdμ = μ(Ω).
A
Proof We may assume 0 < μ(Α) < oo. Put A0 — A and
Rk is the set of points in A with ^(ω) = к. For к ^ 1, Ak is the set of points in Ac
which visit A for the first time at time к.
We have μ(Α1ί) = μ(Α1ί+1) + ju(i?fc+1), (к ^ 0), and, hence,
μ(Αη)= Σ μ№)+Ηιημ(Λ).
к = п + 1 к->оо
00
By the recurrence of τ, μ (Α0) = Σ β №)> which implies lim μ(Α1ί) = 0. Now the
k = l k->oo

20 Measure preserving and null preserving point mappings
assertion of the theorem follows from
00 00 00 00
μ( (J τ~"Α) = Σ β(Α„) = Σ Σ μ№)
η = 0 η = 0 η = 0 к = п + 1
fc=l A
Higher moments of rA have been studied by Blum and Rosenblatt [1967] and by
Wolfowitz [1967].
For conservative null preserving τ, we can define a map τΑ of Aret into A by
ω -> тГа(С0)ω. ω is mapped to the place where it first returns to A. Because of τΑχ Β
00
= (J Якпт~кВ, this map is measurable and null preserving. It maps Ainf into
fc = l
Ainf and one has τΑ = τΑ on Ainf. Neglecting a nullset, we assume A = Ainf. τΑ
is called the transformation induced by τ on A. Obviously, τΑ is conservative.
Theorem 3.7 (Kakutani [1943]). If τ is a conservative endomorphism of
(Ω,^,μ), the induced transformation is an endomorphism of (A, An si, μΑ),
where μΑ is the restriction of μ to A.
Proof Take some measurable В a A, and put B0 = В and
Вк+1 = т~хВкпАс, В'к+1 = т~хВкпА (*£0).
We have
μ(Β) = μ(Β[) + μ(Β1) = μ(Β[) + μ(Β'2) + μ{Β2) = ...
00 00
= Σ μ(Βΰ + lim μ(Βό > Σί*№·
fc = l fc-юо fc = l
But Я* is the set Rknx~kB = RknxAxB. Thus μ(£) ^ μ{τΑχΒ). The theorem
now follows by an application of the following lemma to τΑ. α
Lemma 3.8. If τ is conservative in (Ω, si, μ) and μ(τ~χΑ) ^ μ (A) holds for all
Ae si', then μ is τ-invariant.
Proof Otherwise there exists an A with μ (Α) > μ(τ~χ A). Construct A0, Al9...
Ru R2,... as in the proof of the Kac recurrence theorem. Then μ (Α) > μ{τ~χΑ)
= μ(Α1) + μ(Ρ1)^μ(Ρ1) + μ(Ρ2) + μ(Α2) ... ^ Σ /*(**) =/ι (Λ) yields a
contradiction, α fc_1
Proposition 3.9. If a null preserving transformation τ in (Ω, si, μ) is conservative,
and J is the σ-algebra of τ-invariant sets, then J ел A is the σ-algebra of τΑ-
invariant sets. In particular the ergodicity of τ implies that of τΑ.

Recurrence 21
Proof. Ie J means that ω e lis equivalent to τω e I, and, hence, to τ2ω e I, etc.
Therefore ω e Α η lis equivalent to τΑω e A n /, and A n lis τ^-invariant. If, for
some /', An Γ is not τ^-invariant, there exist an coe^ such that сое Г and
τΑ ω φ Γ, от ω φ Γ and τΑ ω β Γ. So the equivalence of the assertions τηω e Γ must
fail for some η and Γ is not τ-invariant. α
Sometimes it is useful to reverse the process of inducing a transformation on a
subset: Let τ be an endomorphism of (Ω, s/, μ) and let Ri9 R2,... be a partition of
Ω, i.e., the R( form a sequence of disjoint measurable sets with union Ω. Let Ω0
00
= Ω, Ωί = (J i?k, and Ω* = {{k, ω): A: ^ 0, ω e Ωλ}, and for measurable
k = i + l
A czQk put μ* ({£} χ yl) = μ (/1). We have defined a measure space consisting of
disjoint copies of the Ω^. Define an endomorphism τ* in Ω* by
τ*ίΑΓωϊ = ί(* + 1'ω) if WeQk+i
K ' } {(0, τω) if ωφΩ,+1.
If we identify Ω with {(0, ω): ω e Ω0}, then τ£ = τ. This is Kakutani's skyscraper
construction. Clearly one can do the same thing with a null preserving τ and come
out with a null preserving τ*. If τ is an automorphism of (Ω, «я/, μ), τ* is again an
automorphism. If τ is nonsingular, τ* is. Clearly τ* is ergodic if and only if τ is
ergodic. Therefore the skyscraper construction is a very simple way of
constructing ergodic automorphisms in non atomic infinite measure spaces. Simply start
with an ergodic automorphism, say, in the unit interval, and choose the Rk with
00
Σ к- μ(Rk) = oo.
fc = l
Notes
In his famous book, Hopf [1937] described the decomposition Ω = С и D for invertible
transformations, and, in 1954, for the much more general situation of a positive
contraction in Lx; see section 3.1. But concepts like wandering set or induced transformation have
no equally simple and intuitive extensions to the operator case. Sucheston [1957], Helm-
berg [1965], [1965a], [1966], Simons [1965], [1971], Wright [1961], Roos [1964],
Tsurumi [1958], Choksi [1961], Helmberg and Simons [1969], and others have therefore
studied the case of non invertible null preserving transformations. One can split D further
into the set of points in D whose orbit enter С and the rest, see Kopf [1982], [1978].
Call two measure spaces isomorphic mod 0 if - after deleting nullsets N and N' from Ω
and Ω'-there exists a bijective map φ: Ω \ N -► Ω' \ Ν' such that φ and φ_1 are measurable
and μοφ'1 = μ'. Endomorphisms τ and τ' in Ω and Ω' are called isomorphic mod 0 if such
a φ can be found with τ'°<ρ = φη; the sets N,N' shall be such that Ω\Ν and Ω'\Ν'
are invariant under τ and τ' respectively. Two dissipative automorphisms τ and τ' in
σ-finite spaces (Ω, л/, μ) and (Ω', я/', μ') are isomorphic mod 0 if and only if the measure
spaces (W0, W0njtf, μ^0) and (Wo, Wq η л?', μ^ό) are isomorphic modO, where
W0, W0 are the sets with Ω = (J τ* W0, Ω' = (J τ'* Wq constructed in theorem 3.2
4-00
k =

22 Measure preserving and null preserving point mappings
In particular the set WQ is determined uniquely up to measure theoretic isomorphism
mod 0; see Krengel [1968/69: part I], where also dissipative flows are classified by
showing that they are isomorphic to a flow ts: (ω0, i) -> (ω0, t + s) in a space W0 χ R, with a WQ
which is uniquely determined mod 0. The recurrence properties of nonsingular flows and
null preserving semiflows were discussed in part II of this paper. Helmberg [1969] has
derived a number of results on the mean recurrence time of measure preserving flows and
semiflows, thereby giving a subtle continuous time version of the theorem of Kac.
If τ is an automorphism of a probability space (Ω, s#, μ), and we put, for some A e $4,
νΑ{ω) = rA((o) (ω e Α), νΑ(ω) = 0 (ω e Ac), then τνΑ is again an automorphism. Neveu
[1969] has studied the question, when, for a random variable ν: Ω -» Z+, τν is again an
automorphism. It turns out that ν must be such that τν is obtained by iterated
compositions of induced transformations on a decreasing sequence of subsets of Ω.
Related results for more general groups of automorphisms were given by Geman and
Horowitz [1975].
Jacobs [1967] has shown that the assertion of the Poincare recurrence theorem remains
valid if one replaces the stationarity assumption μ = μ°τ~1Ъy a. recurrence property for
the measure: Let μ be a probability measure on a complete, separable, metric space and let
τ: Ω -► Ω be continuous. If the orbit μ ° τ-1, μ ° τ~ 2,... of μ returns into each
neighborhood of μ (in the weak topology given by μη -> μ if \/άμη -* J/i/μ for all bounded
continuous/), then the orbit τω, τ2 ω,... of μ-a.e. ω returns into each neighborhood of ω
infinitely often.
Kurth [1975] considered homeomorphisms τ of a topological space Ω with countable
basis and with a finite invariant measure, and gave a decomposition related to Poincares
theorem into departing points, asymptotic points and recurrent points.
Barone and Bhaskara Rao [1981] studied the recurrence theorem for finitely additive
measures on a σ-algebra. Then a.e. point is k times recurrent for fixed k, but not
necessarily infinitely recurrent.
The following result of Khintchine [1934] is proved in many books (and never applied):
If τ is an automorphism of a probability space, then for ε > 0 and В е jtf there exists L such
that any interval of length L contains at least one k with μ(τΗΒηΒ)^ μ(Β)2 — ε.
D. Maharam [1964] has "extended" nonsingular transformations τ of Ω to endomor-
phisms τ of Ω χ U+ mapping the fiber of ω onto the fiber of τω. τ is conservative iff τ is
conservative.
§1.4 Shift transformations and stationary processes
1. Canonical representation of processes. We now discuss a class of examples
which is of great importance in probability theory.
Let (E, ^) be a measurable space and / Φ 0 an arbitrary index set. The
product space EJ is the space of all functions ω-.J-^E. If ω,- is the value the function
ω takes iny'e /, the map Χ/, ω -> ω,- is called the j-th coordinate map and Xj((o)
= ω,- they'-th coordinate of ω. In the case /= Z+ we can identify ω with the
unilateral sequence (ω0, ωΐ9 ω2,...)? and in the case J— Ζ with the bilateral
sequence (..., ω_1? ωθ9 ω1? ω2?...).
An £"-valued random variable Ζ defined on an abstract probability space
(Ω'? s/\ P) is nothing but a measurable map Ζ: Ω' -* E. A family Υ = {Yj9je J}

Shift transformations and stationary processes 23
of E-valued random variables is called an E-valued stochastic process with
parameter space J. We may write Yj = X} ° Y, when У is considered as a map Ω' -* EJ.
Ε is called state space of У. The product о -algebra 3F J is the smallest σ-algebra in
EJ containing all σ-algebras Xf13F (ye J). 1С can be considered as a random
variable (Ω', s/') -* (£J, JFJ).
The distribution of Ζ is the measure Ρ ο ζ-1 on &, and the distribution of Υ is
the probability measure Ρ^Γ1 on #4 For distinct у1?У2> ...,y„e / and
Ά> = i/i> · · · > ЛЬ ^Jo can be identified with E". The map У}0:
ω -* (yj4 (ω),..., ^·η(ω)) *s an ^"-valued random variable (for the σ-algebra 3Fn
= ^Jo). The distribution of J}0 is called the η-dimensional marginal distribution
corresponding to {jl9... Jn}.
The distribution of Υ is uniquely determined by the finitedimensional marginal
distributions: P° Y~x is the unique measure on 3F3 such that
(Poyi)(f| X^Aj) = P(f] YfiAj)
jeJ0 jeJo
holds for all finite /0 c: / and all choices of Aj e 3F.
If {^-,7*6 /} is a family of probability measures in (E, IF) the product measure μ
= Π μ/ is the unique probability measure in EJ such that
jeJ0 JeJo
holds for all /0 and all choices of A}e #\ The random variables Yj (jeJ) are
called independent if
P(f] YrlA})= UPiYj-'Aj)
holds for all /0, and all Aj. Thus, the random variables Yj (j e J) are independent
iff Ρ ο Υ'1 is the product of the distributions μ,- = Ρ ο ζ.-1.
Now assume / = Ζ or / = Z+. The shift 0: £"J -* £"J is the transformation
defined by Xk (0ω) = Xk+1 (ω). The shift in Ω := Ε1 is often called bilateral shift. It is
bijective and both 0 and 0_1 are measurable with respect toja/ = J^z. The shift in
Ω+ := £"z+ is measurable with respect to sJ — J^z+ and surjective, but not inver-
tible. It is called unilateral shift. (It will be convenient to use some notation like 0,
sf, Xk both in Ω and Ω+. It should always be clear from the context, which
interpretation applies).
If Ε is a topological space, the unilateral shift is continuous in the product
topology, and the bilateral shift is even a homeomorphism.
The simplest way to define a 0-invariant measure in Ω is to take a product
measure. It is easy to see that μ = Ц/е ζ /f/ is 0-invariant iff all μ] are identical.
The automorphism 0 in (Ω, s/, μ) is then called a (bilateral) Bernoulli shift.
Unilateral Bernoulli shifts are defined in the same way in Ω+ with μ = Π/εζ+ /f,· and
identical μ}.

24 Measure preserving and null preserving point mappings
Random variables Yj(jeJ) are called identically distributed if their
distributions μ} agree. Consequently, if Υ = (Ύ0, Yl9 ...) is a sequence of independent
identically distributed random variables, the distribution μ = Ρ ° Y~x of Y is the
the shift invariant measure used for the definition of a Bernoulli shift.
n-l
Kolmogorov's strong law of large numbers asserts that n~x Σ ^converges
k = 0
almost surely (i.e., P-a.e.) to E{Y0) = J Ιο^Λ when the 1^ are real valued,
independent, identically distributed, integrable random variables. Let us see how this
follows from BirkhofFs theorem. We need the ergodicity of Bernoulli shifts and
show a bit more:
An endomorphism τ of a finite measure space (Γ, ЗЙ, v) is called mixing if
(4.1) lim ν(Αητ'ηΒ) = ν(Α)ν(Β)/ν(Σ)
holds for all А.ВеЗЙ. Mixing implies ergodicity because (4.1) cannot hold for A
= В when A is a τ-invariant set with 0 < ν (Α) < ν (Γ). Let si (m, ή) denote the σ-
algebra which is generated by the σ-algebras Zk_13F (m^Lki^n, к ή= ± oo). If θ is
a Bernoulli shift, μ(Αηθ~ηΒ) = μ(Α)μ(Β) holds for all A e si (kl9 k2\
Be si (JuJi) with ^i = ^2 < °° and.A йh < °° as soon asA + л > &2? because μ
is a product measure and 0-n В e si (jx + nj2 + л). Thus
(4.2) lim μ(Αηθ~ηΒ) = μ(Α)μ(Β)
л->оо
holds for all yl, 5 in the union of the σ-algebras si(k, I) with k, Ι Φ ± 00. This
union is dense in si in the metric d(A, A') = μ(ΑΔΑ'). Therefore (4.2) holds for
all A, Be si and θ is mixing. n_x
Now put Kx = {(oeQ + : lim л"1 Σ ^o ° 0k (ω) = £(У0)} and Ky
n_i n->oo fc = 0
= {ω' e Ω': lim η'1 Σ ^(ω') = £(F0)}. Birkhoffs theorem implies μ(Kx) = 1.
n->oo fc = 0
Zk = χ0 ο 0k and Yk = Xk ο Υ yield A, = 7_1 (AJ, so that P{Ky) = (i> о у-1) (/ς)
= μ(Λ^) = 1. But this is the assertion of Kolmogorov's strong law.
Here we have applied the ergodic theorem only to a function on Ω+ depending
on only one coordinate. But we could take any integrable function/: Ω+ -> R,
and then the sequence/0 6k would in general not be a sequence of independent
random variables. Thus, even in the special case of a Bernoulli shift the ergodic
theorem is strictly more informative than the strong law. The main usefulness of
the ergodic theorem, however, is due to the fact that there are many processes Υ
= (Υθ9 Yu ...) for which Ρ ο γ~ι is 0-invariant although the Y0, Yu ... are not
independent.
If Υ = (Y0, Yu ...) is an E-valued process, the distribution μ — Ρ ° Y~x is θ-
invariant iff μ = Ρ ° Υ~ι ο 0_1 = ρ ο (0 ο γ)~ι. Now 0 ° Υ is the process
(Υ1? Υ2,...) for which the time scale is shifted by one. It is therefore natural to
introduce the following definition: An E-valued stochastic process Υ = (Yj9j e J)
with /=Zor/=Z+is called stationary if Υ and Θ°Υ have the same distri-

Shift transformations and stationary processes 25
bution. As the distribution is determined by the probabilities of events of the type
{Yne Al9 ...9 Ytne An}9 we have the equivalent definition: У is called stationary if
(4.3) P(YtieAl9...9YtneAn) = P(Yti+seAl9...9Ytn+seAn)
holds for all n9 all choices of Al9 ..., An e 3F 9 all tl9 ...,/„ e Z + (or Z), and for s
= 1, (and hence for all s). Intuitively, a process is stationary if the random
mechanism generating the process is invariant under a translation of the time scale.
For a process Y= {Yt9 t e J) with /=IRor/= R + , observed continuously,
one can proceed very similarly: For an additive semigroup /we define a family of
shifts 6t in the space EJ by Xs(0tco) = Xt+S(a>)9 where Xs is the 5-th coordinate.
One can then call Υ stationary if the distribution is 0s-invariant for all s e J. This
is the case when (4.3) holds for all ti9..., tn e J and all se J. (This time it is not
sufficient to ask for the validity of (4.3) for a single s9 because U is not a group
generated by a single element).
Let us return to the case of discrete time. By the definition of stationarity a
process is stationary if it generates an endomorphism. Conversely, an endomor-
phism generates many stationary processes:
Proposition 4.1. If τ is an endomorphism of a probability space (Ω, «я/, μ), (Д β')
a measurable space, and f Ω -> Ε measurable, then the sequence (Zf =/nl,
(/ ^ 0)) is a stationary process. Similarly, if τ is an automorphism, (Zi9 (i e Z))
is stationary.
Proof If η e N9 Al9..., An e β and tl9..., tn e Z+ (or Z) are given, we have
μ{ΖίιΕΑ1,...,ΖίηΕΑη) = μϋοτίΈΑ1,...^οτ*ηΕΑη)
= μ(τ-1{/ο^€Αΐ9...9/οΊ*€ΑΛ})
= μ(^τ^1ΕΑΐ9...9^τ^+1ΕΑη)
= μ(Ζίί + 1ΕΑΐ9...9Ζίη + 1βΑη). α
Corollary 4.2. IfY— (Yi9ie J) is an Ε-valued stationary process on (Ω'9 si' 9 P)9
(J = Ζ or Ζ+), andf EJ -+ Ё measurable, then Yi=fo0ioY(ieJ) defines an Ё-
valued stationary process. (ForJ= Z+ this means that Yi=f(Yi9 Yi+l9 Yi+2, ···)
is stationary).
Proof Assume /= Z+. Apply proposition 4.1 with Ω+ instead of Ω, and take τ
= θ and μ — Ρ ο Y~x. The stationarity of Ζ = (Z0, Z1?...) implies that μ ° Ζ"1 is
shift-invariant in Ω+ = Ez\ We have μοΖ'1 = P° y^Z"1 = P°(Zo Г)"1.
But Ζ ο Υ is the process 7= (70, 71?...). The same proof works for /= Ζ. α
For example, if the process Y09 Yl9... is real valued and stationary, the process

26 Measure preserving and null preserving point mappings
i + M-l
^defined by the moving averages Yt = M~l Σ ^ь Μ > 1 fixed, is stationary.
k = i
Clearly, the Yt will in general be dependent even if the Yt are independent.
For the probabilist, the object of primary interest is the stationary process, and
not the measure preserving transformation. Therefore, it is of interest to express
the ergodicity of θ in terms of the process:
If Υ = (Ypj e Z+) is defined on (Ω', si' 9 P)9 call Be si' invariant if there exists
some A e si (in Ω+) such that
(4.4) B={(Yn9Yn+l9Yn+29...)eA}
is true for all η ^ 0. This is equivalent to the existence of an A* e si with В
= Y~x A* and 0-1 A* = A*. One direction of this equivalence ist obvious: If such
an A* exists, we have θ~ηΑ* = A* and В = Υ'1 θ~ηΑ* = {(Yn9 Yn+l9 ...)еА*}
for all η Ξ> 0. If A with (4.4) exists A must be 0-invariant mod μ, because
Υ-1(ΑΑΘ-1Α) = {(Υ0,Υ1,...)εΑ and (7l5 Y2,...) £.4} и {(Г0? Υχ, ···) Μ
and (Fx, У2,...) φ Α} is empty. (A = θ ΧΑ need not hold!) In section 1.1 we have
sketched how to find for A a 0-in variant A2 with μ (AAA2) — 0. It is an exercise to
show that (4.4) implies В — Υ~ίΑ2, so that we can use A2 for A*. The invariant
sets in Ω' for the process inform a σ-algebra«/', and we have J' — Y~x S, where
J is the σ-algebra of 0-invariant sets in Ω+. We could also call a set В' е si'
invariant mod Ρ if there exists an Ae si for which P(B'Δ {(Yn, Yn+U ...) e ^4})
= 0 is true for all n\ but then, again, we could use the same arguments to show
that these are just sets B' which differ from an invariant В on a set of P-measure 0.
Once we have a concept of an invariant set for a stationary process, we also
have a concept of ergodicity. У is called ergodic if any invariant set Be si'
satisfies P(B) = 0 or P(5C) = 0.
Proposition 4.3. If Υ — (F0, Yl9...) is stationary and ergodic and f: Ω+ -> Ё is
measurable, then the process 7= (F0, F1?...) defined by Yt —f(Yb Yi+1,...) is
ergodic.
Proof. We have Ϋ = Ζ ° Y. If 7is not ergodic there exists an invariant 5 for the Ϋ-
process such that 0 < P(B) < 1. For В there exists a shift-invariant set A in Ω+
with 5= 7-1^?. Put A = Z"1^?. It is straightforward to check that the inva-
riance of A under the shift in Ω+ implies the 0-invariance of A. Therefore B
= Y~l A is invariant for the F-process, a contradiction to the ergodicity of Υ. α
BirkhofFs ergodic theorem, spelled out for stationary processes instead of endo-
morphisms τ, now has the following form:
Theorem 4.4. If Y0, Yl9... is a stationary real valued process and Y0 is integrable,
then Нти"1^ Yk = E(Y0\f).
n->oo fc = 0

Shift transformations and stationary processes 27
This follows in the same way from BirkhofFs theorem as Kolmogorov's strong
law.
We have discussed stationarity for a process Y= (YpjeZ+) defined on an
arbitrary probability space. Now note that X = (Xpj e Ζ+) is the identity on Ω+.
Therefore the distribution μ = Ρ ° Υ~χ agrees with the distribution of X. Hence,
all probability statements on У can be expressed in terms of X. For this reason X
is called the canonical representation of processes with distribution μ.
00
2. Remote σ-algebras. The σ-algebra si ^ = f] si (n, со) in Ω+ and in Ω = Еж is
n = 0
called the (right) remote σ-algebra. (Also the term tail σ-algebra is in use). In Ω we
00
can also define the left remote σ-algebra si-«, = f] si{— oo, — ή). si'(m, ή)
n = 0
= Y~xsi(m9 n) is the smallest σ-algebra in Ω' in which the random variables Y}
(/й^;^й,/Ф+ oo) are measurable.
oo oo
s/'a = Π s/'(n,co)(=Y-ls/a>) and */'_„ = Π st'(-co,-η)
n=0 n=0
are the right and left remote σ-algebras of Y. Events like {ω' e Ω':
n-l
lim и-1 Σ ϊ^(ω') ^ α} or {ω' e Ω': Υ* (ω') e Bk for infinitely many k) belong to
n->oo fc = 0
The σ-algebra /infi+ is contained in .я/^ because of yl e </ => Ае si
= si (Q,oo) => A = θ~ηΑβ si(n, oo), (л 2> 0). Similarly У cz «в/^ for processes
with index set Z+.
We call a σ-algebra trivial (with respect to a given measure) if it contains only
nullsets and their complements. Ergodicity means that the σ-algebra of invariant
sets is trivial. Therefore a stationary process Υ = (Yj9je Z+) for which si'^ is
trivial must be ergodic. If ν is a measure on a σ-algebra S% and ^0, 3&γ are sub-σ-
algebras, #0 с #х (mod v) shall mean that for all B0 e &0 there exists a^e^
with viBoABJ = 0.
Proposition 4.5. If(Ypj e Z) is a bilateral stationary process, we have J' с si'^
(mod P) аля? «/ cz si^ (mod μ).
Ргяо/. Because of J a si = si( — oo, + oo) there exists for any AeJ and for any
ε>0 an neeN and an yl£e *$/( — ηε, + oo) with μ(ΑεΑΑ) <s. For all л, yl
= Θ"Μ and μ = μ ο 0"η imply μ(ΑΑΘ~ηΑε) < ε. But 0~Μβ es/(-ne + n, oo),
so that yl can be approximated arbitrarily well by sets in si(m, oo) where m is
arbitrarily large. This implies A e si ^ mod μ. Applying Y'1 also J1 a si'^
mod Ρ follows, α
These observations yield an alternative proof of the ergodicity of Bernoulli shifts.

28 Measure preserving and null preserving point mappings
One simply has to apply Kolmogorov's zero-one-law which asserts that si'^ is
trivial for any process (YjJe Ζ or Z+) consisting of independent random
variables Yj. The following theorem of Blackwell and Freedman [1964] shows that
the triviality of the remote σ-algebra is, in fact, equivalent to an asymptotic
independence condition:
Theorem 4.6. If (YjJeZ+) is any sequence of random variables defined on
(Ω\ si\ P), the remote σ-algebra si'^ is trivial if and only if each Be si' satisfies
(4.5) lim sup | Ρ (Α η Β) - Ρ (Α) Ρ (Β) | = 0.
л->оо Аея#' (η,οο)
Proof For any А е si'^ and any ε > 0 there exists an ηε and an Αε e si (ηε, οο)
with Ρ(Α Α Αε) <ε. As A belongs to each si'(n,oo) we can apply (4.5) to B= Aeto
get Ρ(ΑηΑε) = Ρ(Α)Ρ(Αε). Passing with ε to zero we arrive at Ρ {Α η A)
= Ρ (A)2, so that Ρ {A) e {0,1}. Conversely, suppose si'^ is trivial and fix В e si'.
By the backward martingale convergence theorem (see Doob [1953: thm. 4.2,
382]) the sequence EP(lB\si'(n, oo)) of conditional expectations converges a.e.
and in Li-norm to EP(lB\si'O0), which equals P(B) because of the triviality of
si'^. For any A e si'(n, oo),
\P(AnB)-P(A)P(B)\ = \$(lB-P(B))dP\
A
= \\{EP{UW{n,co))-P{B))dP\
A
u\\EP(\B\si'(n^))-P(B)\\i.
Hence (4.5) follows, α
3. Recurrence times. Let Υ = (YjJ e Z+) be a stationary process and let A be of
the form A = {Ye A} = Y'1 A. For example, A = {Y0 e A0} or A = {Y0 > Yx
+ Y2}· The set A may depend on infinitely many coordinates. We assume
P(A)>0.
Let PA(B') = Ρ(Α)~χΡ(Α'ηΒ') denote the conditional probability of B'
given {Ye A}. PA° Y~x is the normalized restriction of μ = P°Y~X to A.
By the recurrence theorem μ-a.e. point of A belongs to the set^lin/ of points
we A returning to A infinitely often under Θ. We delete the P-nullset
Y~x (A\Ainf) from Ω'. This does not affect the distribution of Y. Then, for all
ω' e A there are infinitely many neN with (1^ (α/), Υη+1 (ω'),...) e A, so that the
random variables i?0 = 0,
Д|+1(ш') = inf {n > RtifoJ. (№'), ζ+1(ω'),...)6 Α), (ι ^ 0)
are finite in A. The recurrence times are given by 7J = Rt — i?r-_l5 (i ^ 1). If rA(d)
= inf {n ^ 1: θηω e A} is the return time studied in section 1.3 and ω — Υ (ω'),
then Riipj') is the time of the i-th return of ω to A. Formally, we have Τ{(ω')

Shift transformations and stationary processes 29
= ΓΑφΚί-ί{ωΊω). Therefore (YRi, YRi+u ...) (a/) is the point θ\(ω\ where ΘΑ is
the transformation induced by the shift on A. Combining Theorem 3.6,
proposition 3.8, corollary 4.2 and proposition 4.3 we have proved:
Theorem 4.7. Let Υ — (Yhi^0)be an Ε-valued stationary process, andletfbe a
measurable map from Ω+ into E. Assume Ae si and μ (A) > 0. Then the process
U^fiY^Y^,...) (i£0)
on (A', A' n si', PA) is stationary. The process (Ub i ^ 0) is ergodic if Υ is ergodic.
In particular the processes T= (Th i ^ 1) and V= (Vh i ^ 0) with Vt= YR. are
stationary, and they are ergodic if Υ is ergodic.
It is clear that an analogous theorem holds for bilateral processes.
4. Bilateral extensions of unilateral processes. In a way, bilateral stationary
processes are simpler than unilateral processes because the bilateral shift is inver-
tible. It is therefore of interest that in all probabilistic questions on stationary
processes we may assume that the process is bilateral if Ε satisfies very mild
regularity assumptions.
Call (£", ^) a Borel space if there exists a bijective map ψ of Ε onto a Borel
subset E' cz U1 which is measurable in both directions. E.g., Polish spaces (i.e.,
complete separable metric spaces) and Borel subsets of such spaces with their
Borel σ-algebra are Borel spaces.
Theorem 4.8. If Υ — (Yjj'eZ+) is a unilateral stationary process defined on a
probability space (Ω\ si\ P) and taking values in a Borel space (£", ^), then there
exists a θ-invariant probability measure μ on Ω = Ε1 such that the distribution of
(XjJ = 0) in (Ω, si, μ) agrees with that of Y.
Proof. For any set A e si (— и, + η) there exists a measurable subset A{n) οΐΕ2η+1
such that A = {сое Ω: (Χ-η(ω), ..., Χη{ω)) e A{n)}. If we put
μ(Α) = Ρ({ω' e Ω': (Ук_„(о/),... ? ^+„(ω/)) e A(n)}\ then the stationarity of Y
implies that this definition is independent of A: as long as к ^ и, and also that the
same number is assigned to μ (A) if A is considered as an element of the σ-algebra
si{— (n + 1), η + 1). Thus these definitions of the marginal distributions of
(Z_„,..., Xn) are consistent. An application of the Daniell-Kolmogoro ν
extension theorem (see e.g. Jacobs [1978]), completes the proof, α
It is a simple consequence of proposition 4.5 that the ergodicity of a unilateral
stationary process is equivalent to the ergodicity of its bilateral extension.
5. Further examples of stationary processes. A class of examples of great interest
in probability theory and in prediction theory is provided by the stationary

30 Measure preserving and null preserving point mappings
Gaussian processes. A real valued process (Ypje Z) is called Gaussian if the
distribution of (Yjt,..., Yjn) is a multivariate normal distribution for arbitrary
jl9..., jn e Z. Such distributions are completely determined by the expectations
E(Yjk) and the covariances Cov (Yjl9 Yjk). In the stationary case the expectations
are identical and the covariances depend only on the differences^ — jk9 so that the
distribution of the entire process is determined by the sequence r{n)
— Cov (Yn, Y0) together with E(Y0). r(n) can be an arbitrary non negative
definite sequence. Small r(ri) corresponds to approximate independence of Yk and
Yk+n. Therefore Gaussian processes are suited for studies of the effect of various
speeds of asymptotic independence. Another advantage is that their finite
dimensional distributions have a density which allows explicit computations involving
only few parameters.
Another important class of stationary processes arises in the theory of Markov
chains. Let Ε Φ 0 be a finite or countably infinite set, and let Y0, Yu ... be a
sequence of E-valued random variables. We can think of Ε as the set of possible
states in which a system can be, and Yn can be considered as the state of the system
at time n. If P(A\B) = P(AnB)/P(B) denotes the conditional probability of A
given jB, the conditional probabilities P(Yn+1 = in+1 \ Y0 = *'о> · · · > Yn — О will in
general depend on the whole sequence /0, il9..., /B+1. We say that F0, Yl9... has
the Markov property and call Y0, Yl9... a Markov chain, if, for all η ^ 1 and all
/0, il9..., in+i e E, this conditional probability is independent of /0, ...,in-x and
just depends on in and iB+1; this is formally expressed by
P\Yn + l — in + l I Yo — *0> ···? Yn — in) = *{Yn + l — ln + X I Yn — ln)·
Intuitively, this means that the probability of a transition from state /„ at time η to
state in+1 at time η + 1 is not influenced by the states in which the system was
before time n. (Strictly speaking Ρ (A | B) is defined for events В of positive
probability, so that P(Yn+1 = /n+il Y0 = *o> ···> Yn — Q таУ be undefined for some
/0,..., /„. But this is no reason to be alarmed because we can then define the
conditional probabilities in accordance with the Markov property).
We say that a Markov chain has stationary transition probabilities if
P(Yn+1 =j\ Yn = i) is independent of n. We then writeptj for this probability of
passing from i toj. Clearly, a Markov chain which is a stationary process must
have stationary transition probabilities.
The theory of stationary Markov chains has many applications and is highly
developed. Many of the results extend to the case of more general state spaces
which need not be countable. The ergodic theoretic aspects are treated in
Chapter 3 and in the supplement. Here we just show how Markov chains lead in a
natural way to stationary processes, and to shift transformations with a σ-finite
infinite invariant measure.
A family (ρ0·: ij e E) is called a stochastic matrix if the/^ are non negative and
Έ]βεΡϊ] = 1 f°r aU i- Given any finite or σ-finite measure ρ = (gt: i e E) on Ε and
any stochastic matrix we can define a measure μρ on Ω+ by

Shift transformations and stationary processes 31
и-1
(4.6) μρ(Χ0 = i0, Xx = il9..., Xn = /„) = ρίο Π At,;t+1·
ί = 0
(As these marginal distributions are consistent, μρ can be extended to jtf).
If π = (π,·) with nt = P(Y0 = i) is the initial distribution of a Markov chain with
stationary transition probabilitiespij9 one easily checks that P° Y~x is the
measure μπ on Ω + . Conversely, if ρ is a probability measure on E, the process
X0,Xl9... on (Ω+, j/, μρ) is a Markov chain with initial distribution ρ and
stationary transition probabilities/^·. σ-finite measures ρ have no interpretation
as probabilities, but come up naturally in the theory of Markov chains.
If μρ is invariant under the shift 0, we have ρ,- = μρ (Χ0 = j) = μρ(θ~χ {Χ0 = j})
= μρ{Χχ =j) for all j. But μρ(Χ1 =j) = ΣίεΕ μρ\Χο = '> zi = /) = Σ,·€ε №·
Call ρ invariant under (#,·) when
(4.7) ZQiPv = Qj UeE)
ieE
holds. For such a ρ, we have for all η and i0, ...,ine Ε
№ρ\Χ1 = *0> -*2 = *1? · * · ? -*n + l == О
= Σ βρ\Λθ = *> -* 1 = *0> · · · 9 Xn + 1 — *n)
n-1
= Σ QiPuo Π Atit+1 = μβ(Χο = ^о? · · ·, χη = О,
ί Г = 0
and this suffices to establish the 0-invariance of μρ.
We have proved:
Proposition 4.9. Tfte «sAi/ϊ in (Ω+, .я/, μρ) defined with the help of a measure ρ on Ε
and a stochastic matrix (p^) is measure preserving if and only if ρ is invariant under
(Pij)-
A measure preserving shift in (Ω+, s/, μρ) is called a (unilateral) Markov shift.
Observe that the procedure described in the proof of theorem 4.8 also permits to
extend σ-finite shift invariant measures from Ω+ to Ez = Ω. With this extension
of μρ the shift in Ω is called a bilateral Markov shift.
If π is any initial distribution and ρ a (ρυ) - invariant measure on Ε with π <^ ρ,
then μπ <^ μρ. Therefore any assertion proved for /ie-a.e. ω e Ω+ with the help of
the 0-invariance of μρ will be true for μπ -a.e. ω e Ω+ even when π is not invariant
under (p^). In this way ergodic theoretic results on endomorphisms in σ-finite
measure spaces can lead to results on Markov chains (Y0, Yl9...) defined on a
probability space.
A state i is called recurrent, if the probability of a return to i given a start in / is
1, and transient otherwise. If all states are recurrent, there exists a σ-finite strictly
positive invariant ρ and the shift is conservative. If all states are transient and
such a ρ exists, 0 is dissipative; see the Notes of §3.1.

32 Measure preserving and null preserving point mappings
6. Stationarity in the wide sense. If Υ = (Yh ie Z+) is a real or complex valued
stochastic process on (Ω', st\ P) there is a different notion of stationarity which
can be used to apply the ergodic theorem of von Neumann. У is called stationary
in the wide sense if all Yt belong to £2(Ω', st\ P) and E(Yt Yj) = E(Y0 7S_^ is true
for all i,j eZ+ with i rg j. Here a is the complex conjugate of a. Recall that </, g>
= E(fg) = \fgdP is the scalar product in L2. If i^is stationary in the wide sense,
and V= Σ ^ϊί put SV= Σ ^Υί+1. Then
i = 0 i = 0
Ι|5Κ||2=<Σ«;^+ι, Σ«^+ι>
i = 0 j=0
= Σ Σ ^jE(Yi+1Tj+1)
i=0j=0
= Σ Σ о^.ВДГ;) = ||V||2.
i = 0 j = 0
m η m
If F= Σ )5fcrk is a different representation of V, \\ Σ Щ Yt- Σ βΜ\ι = 0
Л = 0 i=0 fc = 0
η m
implies || Σ α* ϊί+ι — Σ β* **+ι IL· = 0. Hence S is defined unambiguously. S
i=0 k=0
acts isometrically on the linear span §0 of { Yb ι ^ 0}, and extends to an isometry
S of the Hubert space ξ> — cl9)0. When we apply von Neumann's theorem to S
n-l
and Y0, we see that n~l Σ ^ converges in L2-norm.
i = 0
A stationary process Υ = (Yh i ^ 0) is stationary in the wide sense if Y0 belongs
to L2. Apart from this boundedness condition stationarity in the wide sense is a
much weaker property than stationarity. However, if У is a real valued Gaussian
process which is stationary in the wide sense and satisfies E( J9 = 0 for all /, then
Υ is stationary, because the «-dimensional normal distribution with given
expectation is determined by the covariances.
For bilateral processes stationarity in the wide sense is defined in the same way,
and S is unitary.
7. Asymptotic mean stationarity. Consider a measurable transformation
τ: Ω -> Ω of a probability space (Ω, «я/, μ), μ is called asymptotically mean
stationary (with respect to τ) if
n-l
(4.8) fi(B):=limn-1 Σ μ(τ~ιΒ)
ί = 0
exists for all Be si. Clearly μ is τ-in variant, μ is called the stationary mean of μ.
The Vitali-Hahn-Saks theorem implies that μ is a probability measure. The
following theorem has been proved by Mrs. Dowker [1951] for invertible null
preserving τ, and by Gray and Kieffer [1980] in the general form:

Shift transformations and stationary processes 33
Theorem 4.10. μ is asymptotically mean stationary iff An{x)f converges μ-a.e./br
all bounded measurable f
Proof. If Αη(τ) 1Β converges a.e. for Be si, then (4.8) is obtained by integrating.
For the converse, consider the set Bf of points ω for which Αη(τ)/(ω) converges.
The τ-invariance of Bf implies μ(Β/) = fi(Bf). Birkhoff s theorem yields fi(Bf)
= 1. π
Let us say that μ' asymptotically dominates μ (with respect to τ) if μ'{Β) = 0
implies Ιίπιμ(τ~ηΒ) = 0.
Proposition 4.11. If a τ-invariant probability measure μ! asymptotically
dominates μ, then μ is asymptotically mean stationary.
Proof. BirkhofFs theorem implies μ'(Β/) = 1. Hence Ιίτημ(τ~ηΒ/) = 1. But the
sets T~nBf are identical. Hence μ(Β/) = 1. α
An ^-valued process Υ = (Y0, Yu ...) on a probability space (Ω', sf'9 P) is called
asymptotically mean stationary if Ρ ο γ~χ is asymptotically mean stationary
n-l
under the shift in Ω+. This means that lim л-1 Σ Р((7ь If+ι, У{ + 2,---)еЕ)
i = 0
exists for measurable В с Ω+. Theorem 4.10 asserts that this is equivalent to the
n-l
convergence P-a.e. of и-1 Σ f(Yh Yi+i9 Yi + 2, · · ·) for all bounded measurable/
οηΩ+. ί = 0
The point of these considerations is that asymptotically mean stationary
processes come up in a natural way in several examples. E.g., asymptotic mean
stationarity is stable under conditioning, while stationarity is not.
Notes
Borel [1909] proved the strong law of large numbers for independent Yu Y2,... with
ρ = P(Y( = 1) = 1 — P(Yt = 0). Kolmogorov [1930] proved the almost sure convergence
η
to 0 of я"1 Σ iXi — Ε Yd when Yu Y2,... are independent random variables with
00
Σ<*Γ^γ(ϊΟ<οο
and 0 < an t oo. This is used in the proof of his strong law for independent identically
distributed random variables with finite expectations, of which the first statement seems to
be that in the book of Kolmogorov [1933].
Doob [1953] has given an account of the relations between measure preserving
transformations and stationary processes.
Theorem 4.7 is a generalization of results of Mrs. Moy [1959] and Ryll-Nardzewski

34 Measure preserving and null preserving point mappings
[1961], who proved that the process (7J) of successive time differences between visits to A
is stationary and ergodic under PA, when A is of the form {Y0e B}.
Geman and Horowitz [1976] constructed ergodic stationary proceses (Xn) for which
the return time Ν(ω) = inf {n ^ 1: Χη(ω) = Χ0(ω)} is finite a.e. although X0 has a
continuous distribution.
The recurrence of Sn = Χγ + X2 + · ·. + Xn received a lot of attention recently. Call (S„)
recurrent if Ρ (3n > 1: Sn e U) = 1 for each neighborhood Uof 0. Atkinson [1976] showed
for ergodic (Xn) with integrable Xx that (Sn) is recurrent iff EXг = 0. Berbee [1981]
showed for Revalued stationary processes that Ω is the union of the sets {ω: \\ Sn (ω) || —► oo} and
{ω: each Sm(co) is a limit point of (Sn(co))}. Related references are Dekking [1982], West-
man [1980]. Aaronson-Keane [1982] analysed the asymptotic growth for N -» oo of the
number of visits to 0 of (Sn) up to time N, when (Xt) stems from certain rotations of [0,1 [.
There exists an extensive literature on Gaussian processes; see Neveu [1968],
Ibragimov-Rozanov [1978]. Ergodic and mixing properties of Gaussian processes have
been explored by K. Ito [1944], [1944a], [1952], Maruyama [1949], Grenander [1952],
Leonov [1960], Nawrotzki [1969], Totoki [1964], [1970], Newton [1966], [1968], and
Versik [1962]. For a Gaussian automorphism θ ergodicity, weak mixing and r-fold mixing
are equivalent to r(n) -» 0. θ is isomorphic to a Bernoulli shift iff the spectral measure m
(with Fourier transform (r («)) is absolutely continuous with respect to Lebesgue measure.
The books of Chung [1967], Freedman[1971], and Kemeny-Snell-Knapp [1966] are well
known references on Markov chains.
Wide sense stationary processes are of great interest in prediction theory; see Urbanik
[1967] and Rozanov [1967].
Under various mixing conditions one can derive central limit theorems (and stronger
results) for stationary processes; see Ibragimov-Linnik [1971] and Philipp-Stout [1975].
Stationarity plays an important role in studies of point processes; see the books of
Kerstan-Matthes-Mecke [1974], and Franken-Konig-Arndt-Schmidt [1981], and the
lecture notes of Neveu [1976], and see § 6.2. Rolski [1981] discusses stationary random
processes associated with problems in queuing theory. (He also considers asymptotically
mean stationary processes, calling them stable.)
Ambrose [1941] and Ambrose-Kakutani [1942] have established a basic link between
discrete and continuous time by their representation of flows by flows under a function. If
{τ,: t e Щ is a (proper) measure preserving flow in a probability space (Ω, я/, μ), there
exists a cross-section Ω0αΩ such that each orbit meets Ω0 during a discrete sequence of
times. Papangelou [1974] has connected this to the theory of stationary point processes.
De Sam Lazaro and Meyer [1975] have worked out a whole theory of flows, stationary
processes and point processes. Related work in a probabilistic setting appears also in
papers of Prizva [1972] and Nawrotzki [1982].
Krengel [1969] and Kubo [1969] extended the representation to nonsingular flows, and
Krengel [1969], [1971a] to semiflows and "filtered" flows, where the representation has
to respect additional structure. (These papers were overlooked by de Sam Lazaro and
Meyer, who derived a representation of filtered flows in their setting).
Rudolph [1976] showed that, for ergodic measure preserving flows, Ω0 can be found
such that the times between two visits to Ω0 assume only two values. This was slightly
improved and extended to nonsingular flows by Krengel [1976 a]. Arques and Gabriel
[1977] obtained this for filtered flows.
Kieffer and Rahe [1981] have applied asymptotic mean stationary processes in inform-
n-l
ation theory. They also proved the convergence a.e. of η'1 Σ Xt X2 ... Xn, when (Xt) is

Kingman's subadditive ergodic theorem, the multiplicative ergodic theorem of Oseledec 35
an asymptotically mean stationary process with values in the set of b χ b stochastic
matrices.
In a sense all ergodic automorphisms in probability spaces satisfying mild regularity
conditions can be represented as shifts with finite or countable state space E. This follows
from the theory of generators of measure preserving transformations. We refer to the
surveys of Krieger [1975] and Sujan [1983] and to the book of Denker-Grillenberger-
Sigmund [1976]. Grillenberger and Krengel [1976 a] showed that one can even prescribe,
for fixed n, the distribution of (X0,..., Xn), if the automorphism does not have too large
entropy.
§1.5 Kingman's subadditive ergodic theorem and the
multiplicative ergodic theorem of Oseledec
In 1968 an important new impetus has been given to the study of ergodic
theorems for measure preserving transformations by Kingman's proof of this
subadditive ergodic theorem, which opened up an impressive number of new
applications.
We give a proof of Kingman's pointwise convergence theorem via a maximal
inequality which extends Wieners maximal inequality to the superadditive case.
We then sketch some applications. Following Raghunathan [1979] and Ruelle
[1979] we show that Kingman's theorem can be used to derive the multiplicative
ergodic theorem of Oseledec [1968], which is of considerable interest in the study
of differentiable dynamical systems.
1. Discrete parameter subadditive processes. Let τ be an endomorphism of
(Ω, si, μ). Put Q = {(/, k) e Z2: 0 й i < k}. A family F= {FUk: (/, к) е Q} of inte-
grable functions is called a subadditive process if
(0 ^°τ = ί?+ι,Ηΐ ikhk)eQ)9
(ii) FuuFitk + FKl ((Ц),(к,/)б0,
and
(iii) y(F) = infin-^Fo^dp.neN} > -oo.
Fis a superadditive process if { — Fik\ (/, к) е Q} is subadditive and an additive
process if it is both subadditive and superadditive. An additive process simply
fc-1
has the form Fik = Σ ^ο,ι °τ*· BirkhofFs theorem asserts the a. e.-convergence
j = i
of n~xF0n for additive processes and Kingman's theorem extends this to the
subadditive case. y{F) is called the time constant of the process F.
Put gw = Ji70,n^Ai· As τ is measure preserving, (i) implies |^,кф
= Ji?+i,fc+i άμ. Using (ii) we therefore see that the sequence gl9 g2,... obtained
from a subadditive F is a subadditive sequence of real numbers, i.e., it satisfies
(5.1) gn+mugn + gm, (n,meN).

36 Measure preserving and null preserving point mappings
The following elementary lemma is classical:
Lemma 5.1. If(gn)n^i ^ a subadditive sequence of real numbers then n~1gn con-
verges to inf {n~1gn: ne N} =· y.
Proof. Given N we may write η = knN + rn with 1 ^rn^N and
n~lkn -* N~x{n -* oo). We find that γ й n~xgn й n~x(knNN~xgN + grJ
й N~1gN + ε for large η because n~1grn tends to 0. As N and ε > 0 were arbitrary
the lemma is proved, α
To obtain a maximal inequality it turns out that it is better to change the sign and
to consider superadditive processes. For them the time constant is given by γ (F)
= sup {n~lgn\ neN) = lim n~1gn.
n->oo
We denote the number of elements of a set A by card (A). The following
inequality is due to Akcoglu-Krengel [1981]:
Theorem 5.2.
α > 0, and Ε
If F= {FUk: (i,k)e Q) is a non negative superadditive process,
= {ω: sup (n'1 F0tn(a>): η e N} > a} then μ(β) й α-1 У (Л·
Proof Let TV be a fixed integer, and let
EN = {ω: sup {h-1F0,„(p): 1 й п й Ю > ос}.
Let К > N be another integer, and for each ω e Ω define
Α (ω) = {к: 0 S к < Κ- Ν9 τ* ω e EN}.
Let kx = kx (ω) be the smallest element of Α (ω). There is some ηγ — nx (ω) with 1
й nt rg N and F0ni(Tklco) > ocnv
Let k2 = k2(co) be the smallest element of Α (ω) with k2 ^ k1 + nv Find 1
^n2 = η2(ω) rg N with F0tn2(rk2(o) > ocn2. Continuing in this way one finds
finitely many ki9...9kr9ni9...9nr such that Α (ω) is contained in the union of the
disjoint intervals [ki9 kt + щ\_ and FQtni(xkiai) > ocni9 (i = 1,..., r = r (ω)). The
superadditivity and nonnegativity of F and the fact that all these intervals are
contained in [0, AT[, yield
r
^ο,κ(ω)^ Σ^,,*ί+η,(ω)
i = l
= Σ Fo,nA*k,0)) ^ α Σ Щ ^ α card (Л(ω)).
i=l i = l
Integrating over Ω and noticing that
J card (Α(ω))άμ = J * Σ * 1ΕιΛτ»ω)</μ = (Κ-Ν)μ(ΕΝ)
k = 0

Kingman's subadditive ergodic theorem, the multiplicative ergodic theorem of Oseledec 37
we obtain &.μ(ΕΝ) (Κ—Ν) <= gK. Dividing by К and letting K-+00, μ(ΕΝ)
<; a~xy(F) results. Then let N tend to 00. α
Theorem 5.3 (Kingman). Let τ be an endomorphism of a measure space (Ω, «я/, μ),
and F = [Fuk: (/, k) e Q} a subadditive process for τ. Then n~x F0tH converges a.e.
to a τ-invariant integrable limitJ. If μ is finite, n~l F0n converges also in Lx-norm to
?, and y(F) = \3*άμ holds.
Proof Passing to { — Fik) we may assume that Fis superadditive. Subtracting the
fc-1
additive process F{tk= Σ ^ο,ι°τ7? f°r which the assertion follows from
j = i
BirkhofFs theorem, we may further assume Fik ^ 0.
Let/and/be, respectively, the pointwise lim sup and liminf of n~1F0n as
η -* oo. We know that/is integrable because of Lemma 5.1 and Fatou's lemma. If
m ^ 1 is a fixed integer then/and/are also the lim sup and lim inf of (km)-1 F0km
as к -> oo, because F0tn is monotonely increasing in n.
Let α > 0 be given and let Ε = {ω:/(ω) —/(ω) > α}. We now show μ(Ε) = 0.
Let ε > 0 be given. By lemma 5.1 there exists an integer m with m~1gm>y (F) — s.
1-1
The process Hm = {Щг} with Hknl = Σ Fo m° tmj is additive, and the process Fm
j = k
= {F^tl} with F^j = Fkmlm — Щх is superadditive and non negative. We have
y(Fm) = my{F) - у^Н™)^ my (F) -gmu sm. By Birkhoff s theorem к'1 Щл
converges a.e.. Therefore
/-/^sup-i-FoV
jk^i km
We now apply theorem 5.2 and obtain
μ(Ε) ^ /i(sup {к~1Е^л\ к^1}>ат)й (am)_1y(Fw) й ос'1 ε.
We have proved the a.e.-convergence of n~lF0fII to an integrable/=/.
If Am is the limit of (nm)'1 Щп we can use the im-invariance of hm and the
estimate 0 uf- hm й sup {(km)11 F£k\ к ^ 1} to conclude that
/*(l/-/° тж| > 2a) ^ /i(/- hm > a) + /i(/° tw - Am > a) < —.
If m was large enough (m + l)~1gm+1> y(F) — ε holds as well, and the same
argument gives us μ(|/—/° Tm+11 > 2α) ^ 2ε/α. Combining these estimates we
see that /ι(|/-/°τ| > 4α) = ju(|/°tw -/°tw+1| >4α) <4ε/α. As ε > 0 and
α > 0 were arbitrary the τ-invariance of/follows.
If μ is finite the limits 0 rg Am exist also in the Z^-sense, and the integral of hm
equals ra_1gm rg yO?7)· The superadditivity of i7 implies A2m ^ Am. Therefore the
sequence A2' is increasing and tends in Lx to a limit /f° with \Η°άμ = у (F). Given

38 Measure preserving and null preserving point mappings
/)>0we may fix an m of the form 2* such that \\И° — Am||1 < η and \m~1gm
— y(/OI<»f-
Writing n ^ m in the form n = knm + rn with 0^r„<mwe observe that
0 ^ F0,n - Я£кп ^ Fo,(kn+i)m - Щ,кп- This implies
ί (^0,n ~ Щ,0^ = S(*„ + l)m - ^nSm ^ (^n + 1)™У (i7) ~ fc„grf
^ knmr\ + my(F).
Now n~1(knm) -> land
+ ||(A:nm)-1i^,kn-A-||1 + Pm-A00||1
yield the Lx-convergence of л"1/^ to A00 =/. α
The assertion of a.e.-convergence in theorem 5.3 can be proved under slightly
weaker assumptions when μ is finite:
Theorem 5.4. Let τ be an endomorphism of a finite measure space (Ω, «я/, μ) and
F — Ш,к: {U к) e Q} a family of real valued measurable functions satisfying (i), (ii)
and ||^ο,ι+ Hi < oo. Then n~xF0 n convergences a.e. to a τ-invariant limit f™ with
Jjr^ = inf{J«-1F0>n^:«^l}.
(ll^o,i+ Hi < °° implies ||^,fe+ Hi < oo for all (/, к) е Q; therefore the integrals
J^k(ijU are well defined, although we may have ||i?,jTlli = oo.)
Proof It suffices to consider the case inf {J л-1 Ρ0ηάμ) — — oo. The processes FN
defined by F*k = Max (Fitk, — N(k — i)) satisfy all three properties (i)-(iii) for all
N ^ 1. Therefore the limits/^ = lim„_> т n _1 Fqh exist a.e. and in Lx. Let/30 be the
limit of the decreasing sequence/1,/2,... It is easy to check n~x F0n ->/°°. For all
TV, \Γ<Ιμ й ί/ΝΦ = y(FN) = inf {J/i"1/^^/!} й J л_1<пФ· Letting N -* oo
it follows that \Γ°άμ ^ \n~l Ρ0ηάμ for all n. D
We remark that it is possible to construct a process Ffor an endomorphism of a σ-
finite measure space such that (i), (ii), and ||F0tl* \\x < oo are satisfied, but и-1 F0tH
does not converge a.e..
The case μ (Ω) — 1 of the subadditive ergodic theorem admits also a probabilistic
formulation: Notice that by (i) and the τ-invariance of μ the joint distribution of
the random variables {Fik: (/, к) е Q} is the same as that of the random variables
{Fi+ltk+1:(i,k)eQ}. In other words: The family {Fik} of random variables
satisfies
(i*) ^1Д1ег1;...,^еЩ
= /^№i + i,ki + ie5i> "^Fis+ltks+1e Bs)
for all seN, all (il9 кх),..., (iS9 ks) e Q and all Borel sets Bl9..., Bs.

Kingman's subadditive ergodic theorem, the multiplicative ergodic theorem of Oseledec 39
We say that a family F — {Fik: (/, к) е Q} of integrable real valued random
variables defined on a probability space (Ω, л/, μ) is a probabilistic subadditive
process if (i*), (ii), (iii) hold. Given such a process we can construct a shift on a
slightly more complicated space than in section 1.4 as follows: Let Ω' = RQ and
let X(itk) for (/, k) e Q be the coordinate variables. Let Fitk be the restriction of the
coordinate variable X(itk) ta the set
U = {ω e Ω': Χ{ια){ώ) й ДГ((Д)(ш) + Χ(Μ)(ω) ((ι, к), (к, I) e Q)}.
Let s& be the restriction of the product-a-algebra in UQ to Ω, and let μ on
j^ be the measure with
Let θ: Ω -* Ω be the shift defined by ΡίΛ(θω) = Fi+ltk+1 (ω). Then θ is an endo-
morphism of (Ω, j/, Д), F= {Дк} is a subadditive process for 0, and the joint
distribution of {Fuk} under μ is the same as that of {Fik} under μ. Thus, any result
for subadditive processes immediately implies the same result for probabilistic
subadditive processes. We can now drop the term "probabilistic" again and need
not distinguish these notions.
We now discuss some examples of applications:
(a) Percolation. The notion of a subadditive process as studied here is due to
Kingman. Hammersley and Welsh [1965] had introduced a similar notion in
their study of percolation processes, requesting (i*) only for s = 1. There is now a
vast literature on percolation; see Kesten [1982]. We just give a typical example.
Let S = Z2 and let Jf be the family of pairs (w, v) e S2 which are neighbors,
i.e., which have euclidean distance 1. Assume that for any (w, v) e Jf there is a
positive random variable Tu v representing the time needed to travel from и to v. A
sequence u0, uu ..., ur e S is called a path from и to w if u0 = w, ur = w and
(wf_1? щ) e Jf for / = 1,..., r. For u, w e S, the travel time U(u, w) along the
fastest route from и to w is the infimum over all pathes from и to w of the sums
r
Σ Ти._иЩ. It is easy to see that the process Fith = £/((/,0), (A:, 0)) always satisfies
i = l
(ii). If the random variables Tuv are integrable, independent and identically
distributed it also satisfies (i*) and (iii).
(b) The range of a random walk. Let Xl9 X2,... be independent identically
distributed random variables taking values in a topological group C?, and let Sn
η
= Π %i be the corresponding random walk. i?i>k(ω) = card ({^(ω): i <j S к})
ί = 1
is the number of distinct points of G visited in the time interval [/ + 1, k~]. The
process {Ritk: (/, к) е Q] is subadditive. The subadditive ergodic theorem in this
case generalizes a result of Kesten, Spitzer and Whitman treated in the book of
Spitzer [1964].

40 Measure preserving and null preserving point mappings
(c) Random products in Banach algebras. If we convert the multiplicative
inequality II^^H^MJIH^II va^ *n a Banach algebra into an additive one by
taking logarithms we obtain
Theorem 5.5. Let τ be an endomorphism of a finite measure space (Ω, si, μ), and
let Τ: Ω -> 93 be a measurable map of Ω into a Banach algebra 93 for which
fc-1
log+||r(-)|| is integrable. Put ΤίΛ{ω) = Π Τ(τνω). Then there exists α τ-
v = i
invariant measurable function χ: Ω -► R и { — oo} with χ+ e Lx such that
lim n~l log || Γ0,„(ω)|| = χ (ω) a.e.,
и->оо
and
1^«-1ί^||Γ0,„(ω)|μμ = ίηί{«-1ί^||Γ0,„(ω)||φ}
n->oo η
= \χ(ω)άμ.
The special case where 93 is the algebra of (m x m) matrices endowed with any
matrix norm is a result of Furstenberg and Kesten [I960], which required deep
arguments as long as Kingman's theorem was not available. It will serve as the
basis for the proof of the multiplicative ergodic theorem.
2. Continuous parameter subadditive processes. While the derivation of the
continuous parameter version of BirkhofFs theorem from its discrete counterpart is
very easy, one must be more cautious in the case of additive or subadditive
processes. There are two difficulties which do not appear in the study of
t
\f(xs(o)ds.
0
The first one is a technical problem well known in probability theory. It is due
to the fact that / ranges through an uncountable set and two processes which
agree for each fixed / almost everywhere may then have a different limit
behaviour.
Let {ts, s Ξ> 0} be a measurable measure preserving semiflow in a σ-finite
measure space (Ω, si, μ). Qc = {(s, /)eR2:0^5</} will be the new parameter set.
A family F= {Fst: (s, i) e Qc} of integrable real valued functions Fst is called a
(continuous parameter) subadditive process if the conditions
0)c Fs,t ° τΜ = Fs+Utt+U ((s, t) 6βπ^0),
(ii)c Fs,uuFStt + Ft,u {{s,t\{t,u)eQc),
(iii)c inf {t~l \F0^tdp. / > 0} > -oo
are satisfied, and FiStt)(co) is a measurable map of Qc x Ω -* R with respect to the
product-a-algebra in Qc x Ω.
The following example shows that there exist two subadditive processes F,F' in
this sense for which Fst = Fs[t a.e. is true for every fixed (s,t)eQc,

Kingman's subadditive ergodic theorem, the multiplicative ergodic theorem of Oseledec 41
yetlim^^ t~l F0yt = 0 holds everywhere and t~1FQt diverges everywhere as
/ -* oo: Simply take Ω = [0,1 [ with Lebesgue measure and with addition mod 1
and τ, ω = ω + /. If/: Ω -* R is a function which equals 0 on the irrationals and
is unbounded on the rationals we may use Fst(co) = 0 and Fs[t(co) =/(ω + /)
-Κω + s).
Basically, the difficulty is that measure theory permits taking limits, suprema
etc. only along countable index sets. Usually continuity in (s, /) is too strong a
requirement. A process F is called separable if there exists a countable subset
S c: Qc and a set N e si with μ (TV) = 0 such that for all ω φ Ν and all (s, i) e Qc
/ζ,ί(ω) belongs to the closure of each of the sets {/£>|7(ω): (и, v)e SnU} with
(s, t)e U and U relatively open in Qc.
Separability is a very weak requirement. Moreover, if F' is any given process
there exists a separable process F equivalent to F' in the sense that μ(Ρ8ί Φ Fs[t)
= 0 for all (s, t). This is a variant of a general theorem on processes with a
continuous parameter due to Doob. (See e.g. Borges [1966]). It does not disturb
if F satisfies (i)c and (ii)c only in an a.e.-sense because limits and suprema for F
may be taken along the countable set S: Note that S may be replaced by any
larger countable set, so that we can assume (0, t)e S for (s, t) e S, etc. It is easy to
see that, for ωφΝ, sup {Fst(co): (s, i)e U) = sup {FStt(co): (s, t)e UnS} holds
for all relatively open sets U9 and the same is true for inf. Consequently, the
existence of the limit of /_1ί0>ί(ω) as / -* oo follows from the existence of the
limit as / -> oo with (0, /) e S,
The second difficulty requires a more restrictive condition. Kingman [1973]
has shown that for any positive increasing function Γ (7) there exists a separable
additive process Fona probability space such that, for all ω e Ω, F0ft(cj) as a
function of / has derivatives of all orders and yet lim sup^^ Γ(/)-1 F0t(co) — oo.
Intuitively, what happens is that there are some rare sudden high increases in the
process which are almost instantly followed by decreases of the same size. Thus,
as these excursions are very short, they practically have no influence on the
integrals.
To state a condition which avoids such pathologies one can define the maximal
oscillation on an interval / by
Qj = sup{|/£ti|: s<t\s9tel).
Theorem 5.6. IfF is a subadditive separable process for which Ω[0> Χ] is integrable,
t~1F0ft converges a.e. as t -> oo, and the limit is xs-invariant for all s.
Proof We may assume s = 1. If η is the integer part of t we have
Fo,n+i — Ft,n+i й F0tt й Fo,n + Fn,t>
which implies
F(),n + 1 ~ Ω[η,η + 1] = Λ),ί = Λ),η + *2[η,η + 1]·

42 Measure preserving and null preserving point mappings
By Birkhoff s theorem n~x Ω[ηη+1] = л-1 Ω[0,ΐ] ° τη tends to 0, and the result
follows from theorem 5.3. α
If the condition (ii)c is satisfied everywhere and not only a.e. with exceptional sets
depending on u, s, t, the only way the separability enters now is that it guarantees
the measurability of Ω[0>1]. It can then be replaced by the assumption that ΩΙ is
bounded by an integrable function. The separability has a slightly more
important function when one gives the probabilistic definition of a subadditive process
in which no semiflow is specified. The condition (i)c is then replaced by
(i*)c The joint distribution of {FStt: (s, i) e Qc} is the same as that of
{Fs+rtt+r:(s,t)eQc} for any r.
Now separability ensures that all Ω[ηη+1] are well defined and have the same
distribution. η~χΩ[ηη+1] -> 0 a.e. follows because
00 00
Σ μ(Ωίη9η+1]>εη)= Σ μ(Ω[ο,ΐ]>^)^β"1ίΩ[ο,ΐ]Φ<°°
л=1 л=1
converges for all ε > 0.
Of course no assumption like separability or integrability of Ω[0>1] is needed
when one extends the Z^-norm convergence assertion in theorem 5.3 to
continuous parameters.
3. The multiplicative ergodic theorem of Oseledec. If ω moves along an orbit
ω, τω, τ2 ω,... and a visit in τλω is associated with a linear transformation
described by a matrix Α (τ*ω), then the composition of the transformations
performed until time η — 1 is described by Pn(A, ω) — Α(τη~ίω) · Α(τη~2ω) · · · Α{ω).
The theorem of Furstenberg and Kesten, obtained above as a consequence of
Kingman's theorem, tells us that n~x log ||Ρ„(/ί? ω)|| converges a.e. under the
integrability condition log+ ||^4(-)ll e Lx. The multiplicative ergodic theorem
provides more precise information on the limit behaviour of Pn(A, ω).
For some fixed natural number r, (£ denotes the space W endowed with the
euclidean norm, and Μ = M(r) denotes the set of r χ r matrices with elements
from R. For ^eMwe use the norm \\A\\ = sup {||^w||: ue (£, \\u\\ ^ 1}. A* is
the transposed matrix of A.
Theorem 5.7 (Oseledec). Let τ be an endomorphism of a probability space
(Ω,£#,μ) and A{·) a measurable ηιαρΩ-+Μ, (ω -> Α (ω)), with
log+ Μ(·)ΙΙ e ^i(aO· There exists ax-invariant subset Ω' of Ω with μ(Ω') — 1 such
that for ω e Ω' the following is true:
(i) lim (Ρ* (Α, ω) Ρη(Α, ω))1/2η =: Λ (ω) e Μ exists.
n-> oo
(ii) Let exp λχ (ω) < exp Λ2(ω) < ... < exp λ8(ω) be the distinct eigenvalues of
Λ (ω) in increasing order. ( Their number s — s (ω) may depend οηω,λγ (ω) may be
— go). Let 3ν(ω) be the eigenspace corresponding to expAv(co), mv(co)

Kingman's subadditive ergodic theorem, the multiplicative ergodic theorem of Oseledee 43
= dim (δν(ω)) the multiplicity ofexp Λν(ω), and (5ν(ω) = 5ι (ω) + ... + 3ν(ω),
(50 = {0}. Then we have
1
hm -\og\\Pn{A9<o)u\\ = λν(ω)
n-> oo ft
for ue (5ν(ω)\(5ν_1(ω), (ν = 1, ...,s).
(Hi) The functions ω -> mv(co) and ω -* Λν(ω) are τ-invariant.
1
(7t>J //"τ is ergodic, det (/ί(ω)) = 1, and lim sup J — log ||i^(^45 ω)||έ/μ > 0, /Леи
Ях w negative and λ8 is positive.
Before we start with the proof we recall some auxiliary results from linear and
multilinear algebra, and we discuss the meaning of the assertions.
В e Μ is called positive semidefinite if scalar product <2?w, u} is non negative
for all ue (£. For any A e Μ the matrices A A* and A* A are symmetric and
positive semidefinite and have roots of all orders which again are symmetric and
positive semidefinite. The polar decomposition (see Gantmacher [1958:
§14, 263] says that A may be written in the formal = (AA*)1/2C'= C"(A*A)1/2
with orthogonal matrices C, C" e M. As a symmetric, positive semidefinite В
may be written in the form C-1 DC with С orthogonal and D a diagonal matrix
with non negative entries, we see that any A e Μ can be written in the form A
= CDC with orthogonal matrices С, С. Moreover, we may assume that the
diagonal elements of D = (J0) appear in increasing order, i.e., we have 0 ^ dxl
й d22 й ...,.·. d„. They are the eigenvalues of (A*A)1/2. drr is the norm \\A\\.
Applying the polar decomposition to Pn = Pn(A, ω) we see that Pn is of the form
C'n(P*Pn)1/2. Thus (i) means that, for fixed ω, Pn is essentially a power Λη of a
positive semidefinite matrix Л, followed by an orthogonal transformation. As the
latter is an isometry, the length of the vectors Pnu is of the order || Лпи\\. If и is an
eigenvector of Л for the eigenvalue expAv, we have || Лпи \\ — \\u\\ · exp(«Av). If и
is a linear combination of eigenvectors, the component belonging to the largest
eigenvalue is ultimately dominant, (iv) treats the case where each Α (ω) preserves
the volume in (£. The positivity of lim sup J η _1 log || Pn || άμ guarantees that there
are vectors и for which \\Pnu\\ grows exponentially fast. There must then also be
vectors u' for which ||ΡΠ*/|| decreases exponentially fast. This splitting of the
space into expansive and contractive components is of great importance in the
theory of differentiable dynamical system. The numbers Λν(ω) are called
characteristic exponents', with the multiplicities mv(a>) they constitute the spectrum of
(τ, A{·)) at ω. (5ί (ω) с (52(ω) с ... is called the associated filtration of W.
We need a bit of multilinear algebra: Recall that, for 1 <; к rg r, the fc-fold exterior
power /\k(£ of (£ is the set of all formal expressions Σ ciЩ\ л κί2 ... л uik with
i=l
с(бИ,/иеМ, and ui}e (£, when we compute with the following conventions:

44 Measure preserving and null preserving point mappings
(i) Ui Λ ... Λ (Uj + Uj) Λ ... Λ Uk = Ux Λ ... Λ Uj Λ ... Λ Uk
+ Ux A ... Λ Uj A ... Л Uk;
(U) C(ux A . .. Л Uj A ... Л Uk) = Ux A ... Л CUj A ... A Uk;
(iii) for any permutation (πΐ,..., nk) of (1,..., k):
unl a ... a unk = sign (π) ux a ... a uk.
For more detail, see e.g. Kowalsky [1967].
A scalar product is given in Дк© by
(ux a ... л uk9vx a ... At;k> = det(«wi?t;J»iJ=1>...>k).
In particular, the norm \\их л ... л ик\\ is the square root of the Gram-
determinant det («wi? W/»iJ=1>_>k). Thus || ux л ... л ик\\ is the volume of the k-
dimensional parallelepiped spanned by the vectors tt\ , · · · ? ^t? see Greub [1967:
Ch.VII,§3].
The fc-fold exterior power AAk of a matrix A e Μ is the map /\k& -+ f\k&
given by wx л ... л uk -* yl wx л ... л A uk. As above, write A = CDC with
orthogonal C, C, and 0 S dxl^ ...^drr. Because of (CDuh CDuj} = (Duh Duj},
and the fact that the set of fc-vectors Cwx л Cw2... л C^withHwi л ... л uk\\ S 1
agrees with the set of fc-vectors ux л ... л uk with ||ux л ... л uk\\ S 1? we have
Млк|| = sup {det(«/)Wi?^»u=1,...k): ||Wl л ... л ик\\ й 1}·
Putting wx = (0, ...,0,1)*, u2(0, ...,0,1?0)*,..., we see that
ΜΛΊΙ^ Π du.
r-k<i^r
Using a stepwise reduction of the dimension of the parallelepipeds (formula 7.33
in Greub [1967]), it can be checked that rg holds as well.
Proof of theorem 5.7. \\AAk\\S\\A\\k implies that log+ ||,4лк(-)|| is integrable.
Applying the Birkhoff theorem to the function log+ || A (·) || and the Furstenberg-
Kesten theorem to the maps ω -* А лк(со), (1 ^ к rg r), we see that there is a τ-
invariant set Ω' of full measure such that for all ω e Ω\ the averages
n-l
n~l Σ log4" ||^4(т£са)|| tend to a finite limit, and я"1 log \\Pn(AAk, ω)\\ tends to a
i = 0
limit χ (А:, co)elRu{ — oo} for 1 ^ к rg r- The limits are τ-invariant. We fix ω e Ω'.
There exist orthogonal matrices С„(ш), Οη(ω) and diagonal matrices Dn(co)
= (dn(ij, G>))y with Pn{A, со) = C„(co) £>„(ω) Cn{oj>), and 0 й dn(U U ω)
^ d„(2, 2, ω) rg ... rg Jw(r, r, ω). Mostly we shall suppress ω in the notation and
write Cn,dn{iJ\ etc..
Because of ||Р„(/1лк)|| = Π,·-*<ΐ^4ι(Λ 0 and the monotonicity of the dn(i, i)
in /, there exist — oo rg ρΧ rg ρ2 = · · · = £?r < °° with w_1 l°g 4ι(Λ О -* £?*· We may
split the "interval" {1,2, ...,r} = [l,r+l[nN into finitely many intervals
[iv, /v+i[n^>(v = 1,..., j;/i = l?4+i = r + 1), such that ρ^ = ρ7· iff /and/ belong
to the same interval.

Kingman's subadditive ergodic theorem, the multiplicative ergodic theorem of Oseledec 45
Let CBV be the subspace of (£ spanned by the iv+1 — 1 unit vectors et
= (βib <>2h · · · > <>rd*, (1 й i < J"v+i)· In particular CB0 = {0}, (£s = (£. We want to
show that the subspaces G" = С~х(£х converge to a limit (5V.
n-l
To do this, first note that the convergence of и-1 Σ log+ || ^(τ'ω)|| to a finite
i = 0
limit readily implies that, for any ε> 0,\\Α(τηω)\\ < exp (ηέ) holds for all
sufficiently large n.
Lemma 5.8. Assume ρίν > —go. Given ε > 0 there exists Νε with the following
property: Ifve&l has length \\v\\ = 1, and there are numbers fyeR and vectors
t/e<EJ+1 with
(5.2) !> = !>'+ Σ ^C^e,,
i ^ iv + ι
ίΑβ/ι \bt\< exp {-«(ft - ρίν - ε)} for η ^ Νε.
Proof For large л, logdn(jJ) < (ρίν + ε) и holds for ally<;/v+1 — 1. Using
Pn+1 (Α, ω) = Α (τηω) · Ρ„04, ω) and 1; e €B" this implies
||Ρη+1(Λ,ω)ι>|| <exp(/ie) exp (η(ftv + ε)).
Similarly, for large л
^|^|ехр((л + 1)(а-е)).
As the Qt (i ;> iv+1) lie in a bounded interval and ε may be assumed ^ 1 we may
write exp {{n + 1) (ft — ε)) ^ exp («(ft — 2ε)) for large n. Putting everything
together we obtain \bx\ rg exp (fl(ftv — ft + 4ε)). α
In the case ν = 1, ρχ = — 00, the conclusion in this lemma is that, for any ρ > — 00,
\Ъ{\ ^ exp (ηρ) holds for sufficiently large n. We shall assume ρχ > — oo in the
sequel, because the proof in the case ρχ = — 00 is quite similar.
Let π™ be the orthogonal projection of© into (£™ and π™ the orthogonal
projection of© into the orthogonal complement S™ of (£™. The lemma implies
||π^+1^||^Γ||^||6χρ{-«(ρίσ + ι-ρίν-ε)}
for large η and и e (£", σ ^ v. For A: > 1, we can use the decomposition
u!+fc = i;+kou!+4uvB+ko<+1
= nnv+k о ttvw+1 + nnv+k ο πνη+2 οπνη+1 + nnv+k ο πην + 2 ο <+1
to derive the estimate

46 Measure preserving and null preserving point mappings
(5.3) \\nnv+ku\\urkT\\u\\exp{-(n^j)(Qiv+l-Qiv-8)}
й Кх \\u\\ exp{-n(giv+l- ρίν - ε)},
with a constant K1 independent of и e (£", η and k.
Using similar decompositions into projections on the spaces (£™, S™+1 and the
spaces spanned by C~xeh (ie [*v+i, /v + 2Dj we can show that
Цй;:М| й Σ? г · ||u|| exp (- (и +7) (ρίν + 2 - ρίν - β))
fc-1
+ Σ ^ill«l|exp(-/i(ftv+1-ftv-e))T-exp(-(/i+7)(av+2-ftv+1-e))
й К2\\и\\ exp {-«(ρίν + 2 - ρίν - 2β)}.
Continuing in this way we see that there are constants Кг such that, for all large
fl(sayfl = AQ, ue&t, k^ 1, / = 0, ...,s-v-l,
(5.4) ||Sv-ifii||^^l+1||ii||exp{-ii(eiv+i + l-eiv-.(/+l)6)}.
Redefining TVg we may assume Kx — 1.
By (5.3) each sequence п"+кС~хеь (к = 1,2,...), with i < iv+1, is a Cauchy
sequence. If иis fixed but large, it stays close to C"1 et, and we may assume that
the vectors v* = limk^O0n"+kC~1ei are linearly independent. The space (5V
spanned by v{ (i < /v+1) may be considered as the limit of the spaces (£*. It is
clearly independent of the fixed n.
Our next aim is to study the limit behavior of n~l log ||Ρ„04, (o)v\\ for ν e ©v.
Put/(/) = / for ie [i„ i|+1[, (/ = 1,...,s).
Lemma 5.9. Let В — (b^) be an r χ r-orthogonal matrix. Assume that there are
numbers 0 rg oc1 < oc2 < · · · < ocs with \btj\ ^ a/(o/a/oy ^ftew 2?_1 = (bfj) satisfies
\b$\£(r-i)lamfam.
Proof. If tp is a permutation of {1, 2,..., r) with \p(j) — i, and we write ψ as a
product of cycles, one of the cycles has the form (/, ψ (/), ...,\pq (i) = 7). Using | b(j \
^1 we observe
ΙΠ bkttp(k)\ = Π I^V(o,vh+1(ol ^ α/(ί)/α/ο·)·
fc*j ft = 0
The claim now follows from | det В | = 1 and
*e = (detJ?)-1Z±n^V(jk). π
r
If 5 is the matrix CnC~lk, we have C~le{ = Σ bi}C~lkep and (5.4) with con-

Kingman's subadditive ergodic theorem, the multiplicative ergodic theorem of Oseledee 47
stants Κχ — 1 implies that В satisfies the assumptions of the lemma with och
r
= exp (n(Qih — he)). Because of C~lke{ =- Σ Ь$С~* е}, the lemma now implies
that any ν e (£?+k with ||v|| ^ 1 is of the form ν = υ'η + ν'ή with v'n e <£?, \\v'n\\ й 1,
r
ν'ή= Σ Vj^C^ej, and
j= iv+i
| уЛя I й const exp (- η (ρ,. - ρίν - re)).
As this is true uniformly in к it remains true for ν e (5V.
Now for ι < /v+1
lim sup и-1 log\\Pn(A9(o) Cn_1^|| = lim sup л"1 log </„(/, i) = ρ, ^ ρίν,
and hence lim supn~1log\\Pn(A,co)v'n\\ <; ρίν. Fory^ /v+1,
limsup«-1log||Pn(^^)yJ.wC-1^||^
lim sup η _1 log {const exp (- η (ρ,- - ρίν - rε)) · dn (jj)} = - Qj + Qiv + re + ρ,-.
As ε > 0 was arbitrarily small
limsup/i"1log||PB(^^)t?||^ftv
holds for all ve ©v.
On the other hand, if t;e (£ does not belong to (5V, there is a c> 0 with
|| π"t; || > с for large л. It follows then that
Uminf/T4og||PB(^<»)t>||^ftv+1.
Hence lim л"1 log || J^(^4, со) ι;|| = ρίν holds for ve (5V\(5V_1? (v ^ 1).
By the construction of Dn and C„, C"1 e{ is an eigenvector for (Pn(A)*Pn(A))1/2
with eigenvalue dn(i, 0 and an eigenvector for (Pn(A)* Pn(A))1,2n with eigenvalue
4,(i, О1".
As the spaces spanned by the vectors Cn xeh (ie [iv, /v+i[), are the orthogonal
complements of (£"_! in (£" and converge to the orthogonal complements 5V of
©,,_! in (5V, and as dn{i, i)1/n converges to exp ρί? the matrices (Pn(A)* Pn(A))1/2n
must converge to the matrix Л which has 5V as eigenspace of exp λν = exp&v.
This proves (i) and (ii). The assertion (iii) follows from the τ-invariance of
limn_>O0n~1log\\Pn(AAk,co)\\. It remains to prove (iv). The assumption
det(^l(co)) = 1 implies det Pn(A, ω) = 1 and, hence, ||Ρη(^ί? ω)|| ^ 1. In
particular, iog+Hi^Cd, ω)|| agrees with log||P„(>4, ω)||, and log 11^(^1, ω)|| is inte-
grable. It follows that F*k(ω) := log || Pk_i(A, τ'ω) || is a subadditive process, and
the assumption lim sup J и-1 log ||Ρ„04, ω)\\άμ > 0 means that its time constant
у(^р) is strictly positive. As τ is ergodic, limn~1log\\Pn(A, ω)|| = y(Fp) > Oa.e..
||Ρ„(/1, ω)|| = dn(r, r, ω) and ρ{ = lim n~l \ogdn{i, i) now imply
ρ, = y(Fp) > 0. Next observe that
<4(1,1, ω) · £4(2, 2, ω)... <4(r, r, ω) = det (РИ(Л, ω)) = 1

48 Measure preserving and null preserving point mappings
and €4(1,1) й № 2) й ... й <4(', г) yield dn(r, r)~r й d,(l, 1)й <4(', r)"1^"1).
Hence — τρΓ ^ ρχ ^ — (l/(r — 1))ρ,., and q1 is a finite negative number, α
Notes
Subadditive processes. Kingman's proof of the subadditive ergodic theorem was based on
a decomposition theorem: A subadditive F = {Fik} always can be written as a sum of an
additive process and a non negative subadditive process with time constant 0, see § 3.5.
The present proof of pointwise convergence, which follows Akcoglu-Krengel [1981], has
the advantage that the same maximal inequality can be used to derive also BirkhofFs
theorem (as in theorem 7.3). The argument used to show the invariance of the limit and Z,x-
convergence is taken from Derriennic-Krengel [1981].
There are several other proofs of pointwise convergence for μ(Ω) = 1: Derriennic
[1975] used the following maximal lemma: If E= {liminf«_17r0n< 0}, then
lim и-1 \ΕΡο,ηάμ й 0. Steele [1984] has a proof without explicit use of maximal
inequalities. Smeltzer [1977] used upcrossing inequalities generalizing those of Bishop: For a < b
let WN(w) be the maximum of all n, for which there exist 0 ^ u1 < v1 < ... < un < vn 5Ξ N
with щЪ - ща <Ξ Εη.υ.(ω\ (1 <; г' <Ξ η), and FVuUi + i(co) й Щ+1а- щЪ, (1^/^«-1).
Then
\ψ„άμ£-±-\φ0Λ-α)+άμ.
b — а
This quantitative estimate has independent interest. (The proof appears in the
unpublished thesis written at Yale). Neveu [1983] has the following generalization of the
superadditive maximal inequality: If F is non negative superadditive, ge L\, and Ε
= {sup(7^,, — Sng) > 0}, then \Egdμ 5Ξ y(F). However, his proof of the subadditive
ergodic theorem (for ergodic τ) uses only the special case g == a, the inequality in theorem
5.2.
Kingman gave a subadditive version of the maximal ergodic theorem:
If EN = {supFOfB>0}, then fFO)1rf/i^0.
n^N ' EN
He remarked that this does not suffice to prove pointwise convergence.
The examples of applications given here are a sample from many in the survey of
Kingman [1973]. Other applications can be found in the articles of Hammersley [1974],
Kingman [1976], and Derriennic [1980]; see also Steele [1978], and Smythe-Wierman
[1978].
A subadditive process F on a probability space is called independent, if the random
variables Fmuni, Fm2Tl2,... are independent whenever the intervals [ml9 nx [, [m2, л2[,...
are disjoint. Let 3F (m, oo) be the smallest σ-algebra in which all Fik with m <Ξ i < к < oo are
measurable. The intersection of the SF (m, oo) is the remote σ-algebra of F. If Fis
independent, it is trivial. In particular, independent subadditive processes are ergodic. The
following result of Kesten (contained in Kingman [1976], and in more general form in
Hammersley [1974]) has been useful:
Theorem 5.10. If F is a non negative independent subadditive process with \\F0tl\\2<oo,
then lim \\n~1F0n — y (F) \ \ 2 = 0. The sequence \ \ F0j n \ \ 2 is subadditive. If Vn is the variance of
F0n, we have
Σ 2~2кУ2к<оо.
Ь =1

Kingman's subadditive ergodic theorem, the multiplicative ergodic theorem of Oseledec 49
Derriennic and Krengel [1981] proved that n'1 FQn converges in Lp (1 <p < oo) for any
non negative subadditive process with || F0л \\p < oo. On the other hand, they showed that
there exists a subadditive process with sup ||«_1^0,nll2 < °°> f°r which n~1F0 n fails to
converge in L2. Thus, the condition F0 „ ^ 0 is important. In the same paper it is shown
that the subadditive form of Weyl's theorem fails even for non negative continuous Fik on
[0,1[.
Derriennic [1983a] considered processes for which the condition of subadditivity is
weakened: He showed that n'1 F0n converges in Lx for any process F = {Fik} on a
probability space, satisfying Fuk°x = Fi+ltk+i, inf$n~lF0t„ > — oo, and
f(^o,»+* ~ ^o,n ~ Fn,n+k) + άμ й ск with cjk -* 0.
Almost sure convergence holds if, in addition, F0n+k — F0n — Fnn+k ^Akn" («, к ^ 0),
with {hk} an Lrbounded sequence of functions. This can be used to prove the pointwise
ergodic theorem for information. Moulin Olagnier [1983] proposed a different weakening
of subadditivity, see § 9.2.
Wacker [1983], [1984] characterized the processes which can be decomposed into a
difference of two subadditive processes. He also proved central limit theorems, invariance
principles and other probabilistic results for subadditive processes satisfying suitable
mixing conditions, and for mixing processes which can be approximated by additive
processes. His results apply, e.g., to the range of random walks, and to products of random
matrices, processes previously studied by Jain, Orey, Pruitt, Furstenberg, Kesten, Lange,
and others. First steps towards an abstract treatment were taken by Ishitani [1977].
In the probabilistic formulation of BirkhofFs and Kingman's theorem the condition of
stationarity can be weakened, replacing it by super stationarity: If Pf, Ρ" are probability
measures on a Polish (i.e., complete, separable, metric) space (E endowed with a partial
order relation ^ which is closed (i.e. {(a, b): α 5Ξ b} is closed in (Ε χ (Ε), then Ρ' is called
stochastically larger than Ρ" \ί\φάΡ' ^ \φάΡ" holds for all bounded, increasing,
measurable φ: (Ε -> U. If (Xu X2,...) is a real valued stochastic process on (Ω, s/, P) the
distribution of (Xn, Xn +13 ...) is the measure Pn on (S = RN with Pn (B)
= Ρ ({ω: (Χη(ω), Χπ+1 (ω),...) е В}). The process is called superstationary if P1 is
stochastically larger than P2. Intuitively, a process is superstationary if- compared to
stationarity - there is a tendency towards smaller values. It has been shown by Krengel [1976]
η
that η'1 Σ Xi converges a.e. if (Xu X2,...) is superstationary and Xf integrable. Abid
i = l
[1978] has a common generalization of this result and Kingman's theorem.
Schurger [1983] has results of this type for random convex sets; see §9.4. Hachem
[1981] has maximal inequalities and dominated estimates for superstationary processes
and a continuous parameter form of Abid's result.
Krawczak [1985] has proved upcrossing inequalities for superstationary subadditive
processes. In the continuous parameter case the conditions which suffice for convergence
do not suffice for the integrability of the number of upcrossings.
Liggett [1985] has weakened the assumptions in the subadditive ergodic theorem in yet
another direction.
The multiplicative ergodic theorem. Oseledec [1948] proved the multiplicative ergodic
theorem for invertible transformations and matrices, using a reduction to triangular
matrices. The argument of Raghunathan [1979] works in the noninvertible case, and, in fact,
for matrices with elements from a local field. Some steps omitted in his paper were
sketched in the article of Ruelle [1979], who used these results in the proof of a stable
manifold theorem for diffeomorphisms of a compact manifold. Pesin [1977] derived a
stable manifold theorem a.e. with respect to a smooth invariant measure. Ledrappier

50 Measure preserving and null preserving point mappings
[1981] gave related applications. Also Zakharevich [1978] has a proof of the
multiplicative ergodic theorem.
As pointed out by Ruelle, it is useful to isolate the main part of the proof of theorem 5.7:
If AUA2,... is a sequence of r χ r matrices with lim sup «-1 log||v4J| ^ 0, and Pn
= AnAn^1... Al9 then the existence of lim«_1 log||/>nAfc|| for k = l,...,r implies the
existence of \im(P* Pn)l/2n =. A, and the assertion corresponding to theorem 5.7. (ii).
He also proved a perturbation theorem in the case det (Λ) φ 0: If A\, ...,A'n,... is
another sequence, P'n = A'n,..., A'u η > 0, and sup (||Pn — P'n\\enn) is small enough, then
A' = \im(Pn*Pn)1/2n exists and has the same eigenvalues.
The invertible case. Now let τ be an automorphism, and let the matrices Α (ω) be inver-
tible. Assume that log+||i4(-)|| and log4"|M_1(*)II are integrable. Put Ρ_„(ω)
= Α'1(τ~ηω)... ν4_1(τ_1ω). Then there is a τ-invariant Ω" с Ω with μ(Ω") = 1, and a
measurable splitting Rr = W(1)(co) ® ... θ W(s\w) with
lim «_1 \og\\Pn(<D)u\\ = λν(ω) for 0 Φ we W(v)(cd).
The spaces W(v) are called Oseledec spaces; they need not be orthogonal. We sketch
the construction following Ruelle: Put Pn = Α*(τ-"4"1 ω)... Α*(τ-1 ω) Α*(ω), Pn
= Α*(ω)...Α*(τη~1ω), Pn = Α*'ι(τη'ιω)... Α*'ι(ω). Since the spectrum of Л (ω)
- lim (Ρ* Д)1/2п is τ-invariant, it is also the limit of the spectra of (P*Pn)l/2\ But the
spectrum of P*Pn is the same as that of PnP* = P*Pn. Hence, the spectrum of Л (ω) is the
same as that of Л (ω). In other words, the spectrum of (τ-1, Α*()) is that of (τ, Α()).
Now P* Pn = (P* Pn)'1. Hence, the spectrum of (τ, A* _1 (·)) is obtained by changing the
sign of the spectrum of (τ,Α(·)). The filtration associated with (τ, ν4*-1(·)) is
the one orthogonal to that associated with (τ, Α): (§ν(ω) = (5^_ν(ω). Combining
these observations we see that the spectrum of (τ-1,A'1 n"1) at ω consists
of the numbers — λ8(ω) ... < — λ1 (ω) with multiplicities ms(co),..., m1 (ω).
Let (5_s(co) с ®_(5_1}(ω) с ... с ©^(ω) be the associated filtration. Put Wx((d)
= (5Γ(ω)η©_Γ(ω). It is possible to check (5ν_1(ω)η(5_ν(ω) = {0} and ^v-1 (ω)
+ (5_ν(ω) = W. Then the desired assertions follow. One has Α (ω) Wv(co) = Wv(tco).
Ruelle [1982] has extended the multiplicative ergodic theorem and the other related
results to the case of operators in Hubert space under some compactness condition.
Continuous parameters. Let {ts, s ^ 0} be a measurable measure preserving semiflow in a
probability space (Ω, л/, μ). A map (ω, ή -* Α (ω, t) of Ω χ R+ into M = M(r) is called
cocycle Ίϊ Α(ω, t + s) = Α (τ8ω, t) Α (ω,.?) holds for s, t ^ 0 and ω e Ω. Assume that this
map is measurable and that sup {log+ \\Α(ω, u)\\: 0 ^ и ^ 1} and sup {log+ ||^4(τΜω, 1
— w)||: O^w^l} are integrable. The continuous parameter multiplicative ergodic
theorem asserts the existence of lim,^ (Α(ω, ή*Α(ω, t))1,2t = Λ(ω), and the assertions
analogous to (ii), (iii), (iv). Oseledec deduced such a theorem from the discrete theorem.
One can also adapt the proof of the discrete case. This has been done by Crauel [1981],
who gives applications to linear stochastic differential equations.
§ 1.6 Relatives of the maximal ergodic theorem
In a famous paper, Hardy and Littlewood [1930] have introduced the maximal
function M^f — f^ — sup Αη(τ)/ίον the case where τ is a translation in R. They

Relatives of the maximal ergodic theorem 51
obtained estimates ||/* || < ||/|| for 1 < ρ < oo, and a related estimate for
ρ — I
||/* \\x when/belongs to Zygmund's class L log L consisting of all measurable/,
for which |/| (log |/|)+ is integrable.
We shall now apply the maximal ergodic theorem to derive Wiener's [1939]
dominated ergodic theorem which extends this to the case where τ is a general
measure preserving transformation. We shall also discuss Ornstein's converse of
the LlogL - result for ergodic τ, some identities involving sets like Af
— {ω: Snf{(o) > 0 for all η ^ 1}, and continuous parameter semigroups.
Some of these results will be given for a class of positive contractions in Lv
Those interested only in measure preserving transformations may disregard our
comments on operators and read Tkf as /° xk.
1. Dominated estimates. A linear operator Τ in Lx is called an Lx — L^-
contraction if it is a contraction in Lx and satisfies
\\T\\1^=sup{\\Tf\L:feL1nL^\\f\\O0ul}u^
If Γ is a positive contraction in L1? we may define Tf for measurable/with
f~ e Lx by Tf= lim Thn9 where hn is an increasing sequence in Lx tending to/
a.e.. It is an exercise to show that this definition is unique. In the same way we
may extend Τ to the class of measurable functions/with/+ e Lx using a decreasing
sequence hm tending to /
Hopf s maximal ergodic theorem then implies
{Mnhm + k ^ 0} {Mrthm + k^0)
If we first let к tend to oo and then let m tend to oo we find
{Mnf^0}
and thus see that Hopf's maximal ergodic theorem holds even for all / with
f+eLx. (Also Garsia's proof goes through).
Lemma 6.1. If Τ is a positive Lx — L ^-contraction the following inequalities hold
for f with f+ e Lx and α > 0:
(6.1) w{Mnf^*)u f ίάμ,
{Mnf^0L}
and
(6.2) w(Mnf^2*)u f fdμ.
Proof We apply Hopf's theorem to g —f— a and obtain
J (/-α)φχ£0.
{М„д^0}

52 Measure preserving and null preserving point mappings
As Γ contracts the L^-norm, and as Γ is positive, 7Ί rg 11, and hence {/^ a}
c: {Mnf^ a} cz {Mng ^ 0}. It follows that \{Mnf^<x) (f— ν)άμ Ξ> 0, and this
implies (6.1). {Mnf^ 2a} is contained in {Mn(f— α) ^ α}. Hence μ(Μη/^ 2α) is
bounded by
a"1 J (/-α)φ^α-4ΐ(/-οθΊΐι. □
{Mn(f-a)^a}
We say that two measurable functions Χ, Υ: Ω -* R+ are in a maximal type
relation if
(6.3) αμ(Γ^α)^ J Χάμκαο
holds for all α > 0.
Lemma 6.2. If X,Y are in a maximal type relation and ψ: U+ -> IR+ £y ияи Je-
creasing, right continuous with ψ(0) = 0, then
(6.4) \ψ{Υ)άμ й j№>) J Γ1χρ{άί))μ{άω).
]0,Γ(ω)]
Ргяя/. The function φ defines a measure on ]0, oo[, also denoted by ψ, with
ψ(]0» 0) = ψ(0· If v is Ле distribution of Υ on ]0,oo], i.e., v(]a, b~])
= μ(Υβ ]α, 6]), the left hand side of (6.4) equals
l\^{{y^):y^}W{da)v{dy)= J μ(Υ^α)ψ(άα)
]0,oo[
^ J (a"1 J Χάμ)ψ(άα)
]0,oo[ {У^а}
= J J α~ιΧ{ω)μ{άω)\ρ{άα),
{(ω,α):7(ω)^α>0}
and this is equal to the right hand side of (6.4). α
We can now obtain the desired estimates for the maximal operator
Μ„:/->/* = Mmf= sup {AJ: /i £ 1}.
As usual, we shall use χ log + χ as a short notation for χ (log x)+ for χ > 0, and
put 0 log+ 0 = 0. log ist the natural logarithm.
Theorem 6.3 (Dominated ergodic theorem). If Τ is a positive Lx — L^-
contraction, the following inequalities hold for measurable /^ 0:
(6.5) НЛ^/Н^-^-Ш, (1<P<«>)
/7 — 1
(6.6) ЦЛ4/1К й -^τ (μ(β) + f/log+/^)·
г — 1

Relatives of the maximal ergodic theorem 53
Proof. It suffices to prove the inequalities for Mn instead of M^ and for bounded
functions /.
To prove (6.5) we apply (6.4) to \p{t) = /p(/>0)andto Y= Mnf,X = /Using
t~l\p{dt) = t~1dtp =ptp~2dt we obtain
\\Υ\\1 = !φ{Υ)άμ£!/ΐ!ρί>-2ι!ί](!μ
о
By the Holder inequality this is й г^т ll/llp II YWp'1· The bound for/is a
_P_
p-1
bound for Y. Hence || Υ \\p <oo. We can simplify and obtain || Υ \\p iS ||/||p.
To prove (6.6) we shall need the inequality ^ ~
(6.7) alogbualog+a-ble, (Ой a, 0<b).
(log χ iS χ — 1 implies log χ ■ e iS x, and hence log b :g b/e; this implies (6.7) for a
iS 1. For a> 1, alogi> —aloga = alog(b/a) 5Ξ a ■ bja- e = bje).
We may assume μ (Ω) < oo. We apply (6.4) with the function ψ (t) which equals
t for t ^ 1 and is 0 for ί < 1. With F= M„/and X = fwe obtain
J (У- 1)Ф ^ f ^(П^ й \f- (J Γ1ψ(ώ))άμ
0
= J /·ψ-4ί)άμ^ j /logF^
{7^1} 1 {1^1}
^J/logV^ + ^f^^·
Hence (1 - l/<?) || 1% ^ /ι(β) + J/log+/^ and (6.6) follows, α
Theorem 6.3 is due to Hardy and Littlewood for translations, to Wiener for
measure preserving transformations, and to Dunford-Schwartz [1956] for
operators. Dunford-Schwartz have also shown that Γ need not be positive. This
generalization follows via theorem 4.1.1.
If Z/, L" are subspaces of the space of equivalence classes of measurable
functions, an operator M: L! sf -* Mfe L" mapping Z/ into L" is called sublinear if
I M(fx +/2) | й I Mft | + | Mf21 is true for fl9f2 e L'. Μ is said to be of weak type
(PuPi)if L' = Lvi> L" = Lp2 and
P2
μ(\Μ/\>α)^(€·α-'\\/\\ρι)
holds for all/e LPl with a constant independent of/and α > 0. Μ is said to be of
strong type (pup2) if
\\Mf\\P2uc\\f\\Pl
holds for all/e Lpi with a constant с independent of/. The maximal operator M^

54 Measure preserving and null preserving point mappings
clearly is sublinear. As Tis an L^-contraction, M^ is of strong type (oo, oo) with
constant 1. Lemma 6.1 asserts that M^ is of weak type (1,1). By a special case of
an interpolation theorem of Marcinkiewicz (see e.g. Zygmund [1935, vol. II])
these two facts suffice to prove that M^ is of strong type (/?, p) for 1 < ρ < oo. This
is the main content of (6.5).
Sometimes a variant of the arguments above is of use. Say that Χ, Υ: Ω -* R +
are in a weak maximal type relation if
(6.8) αμ(Υ^α)^2 J Χάμ (α>0).
{2Х^л}
Then the same argument as in the proof of (6.4) shows, that for non decreasing,
right continuous ψ: U+ -> U+ with ψ(0) = 0
2Χ(ω)
(6.9) \ψ{Υ)άμ^2\(Χ(ω) J t~^(dt)) μ(άω).
о
Let us apply this with \p{t) = i(log+/)m_1 and m^2. We have dxpjdt =
(m - 1) (log+ f)w~2 + G°g+ 0W_1> and hence
2Χ(ω) 2Χ(ω)
J Г1тр(Л)^т(^+2Х(ш))т~1 J rxdt = m(}og+2X(a}))m.
о о
Let L logmL denote the class of all measurable functions/: Ω -* R for which
\f\ (log+ \f\)m is integrable. We obtain that, if μ is a finite measure, then
XeL logwL implies YeL logw_1 L. Now note that (6.2) means that X = /and Г
= M^/are in a weak maximal type relation. Together with (6.6) we have proved:
Theorem 6.4. If Τ is a positive Lx — L^-contraction in Lx of a finite measure
space, then fe L logmL implies M^ \f\ e L logm_1 L, (m ^ 1).
2. A converse. If τ is ergodic and μ(Ω) — oo, Af^/cannot be integrable for 0
<; / φ 0, because this would imply that ^„/converges in Lx-norm to the a.e.-limit
0. For ergodic τ and μ{Ω) < oo the following theorem of Ornstein [1971] shows
that the sufficient condition fe L logL is also necessary for the integrability of
/*:
Theorem 6.5. If τ is an ergodic automorphism of a finite measure space (Ω, s/, μ)
andf^ 0, then f* e Lx implies fe L logL.
We adapt an argument of Derriennic [1973]:
Lemma 6.6 (Moy [I960]). If τ is an automorphism of a σ-finite measure space
(Ω, я/, μ), Ae si, τΑ(ω) = inf {n ^ 1: τη(ω) e A} with inf 0 = oo, and
00
A* = (J τ + ίΑ, then g ^ 0 implies
i = 0

Relatives of the maximal ergodic theorem 55
(6.10) ^άμ^ΐ'Ϋ g°τkdμ.
A* A k = 0
Proof. The sets Ak = τ* {ω e A: rA (ω) ^ к + 1}, (1 й к < oo), form a disjoint
partition of A*\A. The identity therefore follows from
га~1 oo oo
J Σ £°τ*<//ι = Jgrf/i+ Σ ί g°^k^ = ^άμ+ Σ $ gdμ. Π
A fc = 0 A fc=l Ап{гл>к} Л fc=l Лк
Lemma 6.7. 7/7Ae automorphism τ is conservative and ergodic,/^ 0, and a > 0 w
so large that A = {/* < a} has positive measure, then
(6.11) J /^£2α/ι(/·*£α).
/Too/. The assumptions imply A* = Ω. For юе^с{/<а} one has
гл((о)-1 τΛ{ω)-1
Σ f(i*a>)u2a(rA(<o)-l)u2a Σ 1,
fc = l fc = l
»"^(ω)-1
because otherwise/* (ω) ^r^co)'1 Σ /(τ*ω) ^ α. Applying lemma 6.6 to g
= /and to g = 1 we find k = 0
JyV/ju = J ГΣ /°τ*<//ι^2α{ΓΣ 1(Ιμ = 2αμ(Αύ). π
Лс Л fc = l Л fc=l
iVtfo/ о/ theorem 6.5. We may assume /^ 1 by passing to /+ 1. Let a0
= inf {α: μ(/* < a) > 0}. Now
/(ω)
\ f log fdμ = J/(co) J α χάαμ{άώ)
ι
00
= J a"1 J /{ω)μ{άώ)άα
OO 00
έ'ία_1( ί ί(ω)μ(άω))ά*^2\μ(ί*^α)ά*
1 {/*^a} «о
ao a0
+ ll/Hi j a-^a £ гИ/*^ + Ц/IU J oT^a < oo. a
1 1
The only instance where we have used the invertibility of τ has been the proof of
Lemma 6.6. Therefore the following generalization of the Kac recurrence
theorem (g = 1) shows that Theorem 6.5 and lemma 6.7 hold also for
endomorphisms:
Proposition 6.8. If τ is a conservative endomorphism of a σ-finite measure space
oo
(Ω, si, μ), then (6.10) holds for A* = \J τ~ιΑ andg^ 0.
i = 0

56 Measure preserving and null preserving point mappings
The case where τ is invertible is a special case of Lemma 6.6 because for
conservative τ the definitions of A* coincide. We postpone the proof of the general case to
section 3.3. (Theorem 6.5 for endomorphisms follows also from the invertible
case by a passage to the bilateral extension).
If τ is a dissipative automorphism and/e L\, the inequality (6.11) holds for all
+ 00
α > 0: Simply observe that Σ /(τ'ω)<οο a.e.. Therefore, for almost all
i = — oo
ωe {/* ^ a}, there exists some к ^ 1 with Ле {/* < a}. This is all that is
needed to apply the argument used in the proof of lemma 6.7.
3. The sign of partial sums. For/e Lx we consider the sets
Af = {ω: Snf((D) > 0 for all η ^ 1}
and
Af = {ω: SJ(co) ^ 0 for all η ^ 1}.
A*, AJ, etc. are defined as A* in proposition 6.8. Put
SJ=MSnf.
The following identities have been observed by Marcus and Petersen [1979] for
invertible ergodic τ. Part of their result follows also from a result of Halasz
[1976], who proved that, for ergodic τ in a finite measure space, \/άμ = 0 implies
μ(Λ/) = 0.
Theorem 6.9. If τ is a conservative endomorphism of a σ-finite measure space
(Ω, stf\ μ) andfe Lx then
(6.12) \/<Ιμ = j ε^άμ = J S„/V//i = l/άμ.
A} Af Af A}
Iflifdti is поп negative for all τ-invariant sets I e stf, the four integrals agree also
with ^άμ.
Proof The second identity in (6.12) is obvious since ^/(ω) = 0 holds on
Af\Af. Let rA = rA((D) be the first return time of ω to A = Af. We claim that
(6.13) S*/M = V/(t1'co), (ωβΑ).
ί = 0
m
As хГл ω belongs to A, all sums Σ f(? ω)> (m ^ га)> are non negative. Therefore,
i = rA
there must be some к with 1 ^ к rg гА(<о) and S^f((D) = Skf((o). For all m ^ 1,
^/(τλω) = Sw+k/(co) - 5Λ/(ω) = Sw+k/(co) - 5φ/(ω) ^ 0. This shows that
τ* ω belongs to yl. As rA was the time of the first return, к = rA, and (6.13) is
proved. The third identity in (6.12) now is a consequence of proposition 6.8. To

Relatives of the maximal ergodic theorem 57
prove the first identity we may assume Ω = AJ, since AJ is τ-invariant. But then
Af agrees with Af.
To prove the last assertion we may assume that A^ is empty. For g = —/the
set {M^g ^ 0} then agrees with Ω and the maximal ergodic theorem implies
\(—/)άμ ^ 0. Together with the assumption the proof is complete. D
It is easy to show by example that (6.12) may fail for dissipative transformations.
Marcus and Petersen derive the following corollary using the ergodic
decomposition of Ω, but is is not necessary to use this deep tool.
Corollary 6.10. If τ is conservative, S*f= sup {Snf: η ^ 1}, and
Bf = {ω: Sn/(co) < 0 for all η ^ 1}, then
Af Bf
Proof. By an exhaustion argument we may split Ω into three invariant sets /+, 71
and I0 such that, for any invariant / of positive measure, \^άμ is positive when
/<=/+, negative when / с /_, and zero when / с I0. (6.13) implies the strict posi-
tivity of S+f on Af. If / is any invariant set with μ(Α/ηΙ) > 0, we obtain
0< J Ξ^μ= J fdμ.
Afnl (Afnl)*
Hence Af is contained mod μ in /+.
Applying theorem 6.9 to 7+ we find J S^fdμ — J /V/μ. By symmetry
J S*fdμ = J /V/μ, and the part /0 does not disturb since J /ί/μ = 0. α
Bf I - Jo
Theorem 6.9 admits some interesting probabilistic conclusions: Consider a
stationary process X09 X1,... taking values in { — 1, 0, +1}. In the product space
representation we may write Xt =/<> τ\ where τ is the shift, μ is the probability
measure P. Sn = Z0 + ... + Xn-t is a "walk" on the integers. On ^y = {Sn Ξ> 1
for all и}, 5^/ is = 1. Therefore, if the process (Xt) is ergodic and EX0 non
negative, the probability that the walk always stays above 0 is EX0.
Even without the condition EX0 Ξ> 0 and ergodicity the corollary shows that
the probability that the walk first returns to 0 from above is the same as the
probability that it first returns to 0 from below. [We have S#f= 1 on Af, and
S*f= - 1 on Bf. Hence P(Sn ^ 1 for all n) - P(Sn й - 1 for alb) = P(X0 = 1)
- ρ(χ0 = — 1). But the two probabilities in question are P(X0 = 1) - P(Sn ^ 1
for all n) and P(X0 = - 1) - P(Sn ^ - 1 for all л).]
4. Continuous parameters. The continuous parameter maximal and dominated
ergodic theorems can be deduced from their discrete counterparts by
approximation. Again we treat the operator case right away.

58 Measure preserving and null preserving point mappings
A family if = {Tt, t > 0} of bounded linear operators in a Banach space X is
called a semigroup if Tt+s = Tt Ts holds for all t, s. if is called strongly continuous
at / if || Tsf— Ttf\\ tends to 0 as s -* / for all/e £, and strongly continuous if it is
strongly continuous at all t. We also consider semigroups if = {Tt, t ^ 0} with
parameter set [0, oo[. We do not request that T0 is the identity, though this is the
main case, if is called locally bounded if sup {|| 7^||: 0 < t rg a} is finite for one and
hence for all a > 0.
If if = {Γί? / > 0} is a locally bounded strongly continuous semigroup of
linear operators in L1 (μ) we may define the integrals
(6.14) S0taf=]Ttfdt, (feLJ,
о
in several essentially equivalent ways: Put/„(/, ω) = T[lnj"for i/n rg / < (/ + \)/n.
For given ε > 0 and te > 0 we may find ηε such that η ^ ηε, t > tz implies \\fn(t, ·)
— Γ,/ΙΙι < ε. Therefore there exists a subsequence лу and a measurable/^(/, ω)
on ]0,oo[x Ω such that/Wv(/, ·) converges in Lx and a.e. to f^{t, ·). The Li-
convergence is uniform in / ^ tz. /^ (/, ·) is a representative of the equivalence
class Ttf. When λ denotes the Lebesgue measure,/^ (7, ω) is (Я х //)integrable on
α
]0, ά] χ Ω for each a < oo, and we may put S0af(co) = jY^, ω) Λ.
о
If we choose another subsequence or other points of subdivision than i/n we
get a function/^ (Λ ω) agreeing with/^ (ω, ί) (Я х //)-а.е. and the corresponding
integrals S'0af mil agree with 5O>a/except on a nullset which is independent of a.
For fixed a, 5O>a/is also the Li-limit of the Riemann sums
Κ(η)/=-ηΣΓαίη/,
Π i = 0
We can now deduce a continuous time version of Hopf s inequality:
Theorem 6.11. Let (Ω, «я/, μ) be a σ-finite measure space and if — {Tt, t ^ 0} a
strongly continuous semigroup of positive contractions in Lx. For α > 0 andfe Lx
put
E(oc) = {ω e Ω: sup {S0f<J/(ω): 0 < a < α} > 0}.
Then
(6.15) f Το/άμ*0.
£(α)
Ргяо/. Let Dn = {A:/2n: A: = 1, 2,...} be the set of positive dyadic rationals of
00
order η and D = (J Dw. Applying Hopf s inequality to the function T0/and the
n = l
operator T2-n we obtain, for aefl„ and m^n, the inequality

Relatives of the maximal ergodic theorem 59
(6.16) J Το/άμ^Ο,
£<™>(α)
where
L(m)(a) = {sup { Σ 1i-mT0f: О й к й a2w} > 0}
ί = 0
= {sup {Щ(2т) T0f: Ο^βύκ,ββ DJ > 0}.
Rp(2m) r0/tends to S0 β T0f= S0pfin Li-norm. Passing to subsequences and
finding a common subsequence by the diagonal procedure the convergence along
the subsequence is also pointwise except on a μ-nullset N0 independent of α e D
and β e D. Because of
sup S0tPf= sup S0tfif,
0<β<α 0<β<α
fieD
it follows that μ (Ε(a) \ E(mi) (a)) -* 0 as m{ -* oo along the subsequence. Again by
the strong continuity of the semigroup, the set {T0f> 0} is contained in E(a) up
to a μ-nullset. (6.15) now follows from (6.16). D
It is now not difficult to proceed as in the beginning of this section to prove a
continuous parameter version of Lemma 6.1. for strongly continuous
semigroups {Tt, t ^ 0} of positive Lx — L^-contractions. A generalization of such a
lemma giving a simultaneous estimate for several functions/M is proved in Vol. I
of Dunford-Schwartz [1958; 690].
For fe Lp, (1 <p < oo), and strongly continuous semigroups {Tu t > 0} of
contractions in Lp, it is possible to define S0tOf essentially as above. The p-th
power oif^it, ω) is (λ χ //)-integrable on [0, α] χ Ω. Therefore,/^ is integrable
a
on each [0, α] χ Η with μ{Η) < oo, so that the integrals jY(/, ω)dt are well
defined and finite for a.e. ω. °
It follows from the Riesz convexity theorem (cf. Dunford-Schwartz [1958])
that Lx — L^-contractions are also contractions in Lp, (1 < ρ < oo). (For positive
operators a short proof is given in section 1.7). Using this fact, it is not hard to see
that a strongly continuous semigroup {Tt, t > 0} of Lx — L^-contractions is also
strongly continuous in Lp.
The following theorem is the continuous parameter version of the dominated
ergodic theorem 6.3. We leave the proof (by discrete approximation) to the
reader:
Theorem 6.12. IfZT — \TU t > 0} is a strongly continuous semigroup of positive
Lx — L ^-contractions in Lx (Ω, s/, μ) and M0fODf= sup {a~x S0af 0 < a < oo},
then the following inequalities hold for measurable /^ 0:
(6.17) ΙΙΜο,^/ΙΙ,^^-ΙΙ/ΊΙ,, (!</»< oo),

60 Measure preserving and null preserving point mappings
and
(6.18) \\M0taof\W й j^y (μ(Ω) + i/log+/^).
A similar theorem appears in Dunford-Schwartz [1958: thm. VIII 7.7]. (The
condition of positivity of Tt can be eliminated by theorem 4.1.1).
We now look at the special case of semigroups {Tt} induced by measure
preserving transformations: Let {xt\ t ^ 0} be a measurable measure preserving semiflow
in (Ω, sf9 μ), and Tt the composition with xt. In order to apply one of the theorems
stated above it remains to show that the measurability implies the strong
continuity of the semigroup.
A mapping t -* F{i) of a measure space (C, #, v) into a Banach space X is
called weakly %!-measurable if, for any h from the dual space £*, the
map / -* (F(t)9 A> is ^-measurable. Fis said to be finitely valuedif it assumes only
finitely many values x} e X, and, for x} Φ 0, the sets {te C: F{t) = x,·} have finite
v-measure. Fis called strongly %!-measurable, if there exists a sequence of finitely
valued functions Fn{i) strongly convergent to F{t) for v-a.e. te C. Fis called v-
almost separably valued if there exists a v-nullset C0 such that {F(t): teC\ C0} is
a separable subset of X. A theorem of Pettis asserts that F is strongly c€-
measurable iff it is weakly ^-measurable and v-almost separably valued; see
Yosida[1974: 131].
We apply it with С = [0, oo [, ^ = ^, and ν = Я, where λ is the Lebesgue measure,
and Μ the family of Lebesgue measurable sets. Let Ее si be a set with finite μ-
measure, and F{t) = 1£ ° %t. The measurability of the semiflow shows that, for all
h = 1Л, (>4е л/), (F(t),h} = μ(Αητ^Ε) depends measurably on i. This is
enough to imply the weak ^-measurability of F. Let si be the completion of
& (x) si with respect to Я χ μ. For any set G e si there exists a countably
generated sub-a-algebra sJG of .я/ such that G belongs even to the completion of
@t ® siG\ (the family of sets G with this property is a σ-algebra). Now take
G = {(/, ω): ^coei?}.
^(/) belongs for almost all t to the separable space Lx (Ω, sJG, μ). By the Pettis
theorem F is strongly ^-measurable.
By a theorem of TV. Dunford (Dunford-Schwartz [1958], 8.1.3) a strongly c€-
measurable family F(t) = Ttf Tor a semigroup of bounded linear operators Γ, is
strongly continuous at / > 0, i.e., we have || Tsf— Ttf\\ -* 0 as s -* / > 0.
Approximating a general/e Lx with simple functions we conclude that our
semigroup given by Ttf = fo xt is strongly continuous at / > 0. As the Tt are isome-
tries, we find, for 0 < r -* 0, that || Trf-f\\x = || Tr+tf- Ttf\\x -* 0, and the
semigroup is also continuous at / = 0. Modulo the results of Pettis and Dunford
we have proved
Theorem 6.13. If' {xt\ t ^ 0} is a measurable measure preserving semiflow in α σ-

Relatives of the maximal ergodic theorem 61
finite measure space (Ω, «я/, μ), the semigroup {Tut^i 0} defined by Ttf = f° xt is
strongly continuous in each Lp? (1 ^p < oo).
Notes
A dominated estimate similar to (6.6) was given by Marcinkiewicz and Zygmund [1937]
in the case of independent identically distributed random variables, called the Li.d. case in
the sequel. Wiener's larger constant in the estimate (6.5) was improved by Fukamiya
[1940]. Notes concerning theorem 6.4 are given in section 6.1.
Ornstein's converse of the L log L-result was already obtained by Stein [1969] for
translations in Ud. It was also motivated by a theorem of Burkholder [1962], saying that,
in the i.i.d. case Xi=f°zi, the integrability of/**:=sup„|v4„/| implies fe L log L.
R.L.Jones [1976] has shown that this stronger converse requires the i.i.d. assumption.
B. Davis [1982] has characterized the distributions of/Tor which/** can be integrable for
an ergodic stationary process:
χ 1
Theorem 6.14. Let g be поп decreasing in [0,1] and Μ (χ) = J g(t)dt + J g(t)dt. There
0 1-х
exists a stationary ergodic sequencef'° τ1 such that the distribution off agrees with that ofg
1/2
(under Lebesgue measure) andf** is integrable, iff J \M(x)/x\dx < oo.
о
For g ^ 0 this reduces to the L log L-criterion.
Derriennic [1973] proved a dominated estimate and a converse for ergodic ratios: Let
τ be an ergodic conservative endomorphism of (Ω,^,μ), fgeL\ and g>0 a.e..
Put s = supn(Snf/Sng). If {/log4" (//g)i//i < oo, then {gs^/i < oo. Conversely, if
§g[s + s ο τ]άμ < oo then J/log* (f/g)^ < oo.
Other related work was done by Gundy [1963] and R.L. Jones [1977]. B. Davis [1971]
proved another strong converse in the i.i.d. case: Put/= sup {η _1/°τ"}· As/is dominated
by M^l/I, it is clear that/is integrable for/e L log L. Davis showed that for/^ 0 the
integrability of /implies /g L log L. Krengel and Sucheston [1978] gave an example
showing that this does not extend to the general ergodic case.
Petersen [1979] proved a continuous parameter form of Ornstein's converse. In the
continuous parameter setting the maximal inequality frequently holds even with equality:
Theorem 6.15 (Marcus and Petersen [1979]). If{it, teU} is a measurable ergodic
measure preserving flow in a probability space, feL1, and α ^ |/*/μ, then
(6.19) J fdμ = ocμ(M0,00f>a).
{Mo,oo/>a}
They obtain this from continuous parameter results related to theorem 6.9; see also
Petersen [1983]. (The fact that there is equality in (6.19) was known for translation in U; see
Hewitt-Stromberg [1969:423]. Sato [1982] has proved maximal equalities for semiflows.
Recently, Sato [1984a] obtained a generalization of theorem 6.14, which concerns the
ratio maximal function for ergodic flows. Fefferman and Stein [1971] have maximal
estimates for sequences of functions which unify the Hardy-Littlewood estimates with
results of Marcinkiewicz and Carleson. Bru and Heinich [1981] studied dominated
estimates in Orlicz spaces.

62 Measure preserving and null preserving point mappings
Other dominated estimates. Krengel, Rottger and Wacker [1983] proved: If Y0, Yu ... is a
stationary sequence of positive valued random variables and Ζ = inf„ (F0 + Yl
+ ... Yn-dln, then E(l/Z) й 4£(1/F0). If Nt = sup {n ^ 1: Y0 + ... + Yn^ й '}, then
E(sup NJt) ^ E(l/Z). This allows to prove the Z^-norm convergence of NJt in the case
E(l/Y0) < oo. The integrability condition E(l/Y0) < oo is in a sense optimal.
Motivated by analogies in martingale theory, several authors considered the ergodic
square function
5β(Γ)/=(ΣΜ„/-Λ-ι/Ι2)1/2
1
(with A0f= 0). Jones [1977] showed for ergodic automorphisms τ that SQ(T) is a
bounded operator in Lp for 1 < ρ < oo, and of weak type (1,1). De la Torre [1979] proved
that the boundedness in Lp remains true for positive contractions Tin Lp. Further
generalizations were give by Yoshimoto [1980].
Sometimes one is interested in a τ preserving a measure μ, and in dominated estimates in
Lp(v) where ν has a density w = ά\\άμ. For this we refer to Muckenhoupt [1972], Coif-
man and FerTerman [1974], Atencia and de la Torre [1982], de la Torre [1982], and the
references given there.
This is closely related to the existence of the ergodic Hilbert transform. Cotlar [1955]
obtained an ergodic generalization of the classical Hilbert transform (for translations) as
follows:
η
Theorem 6.16. Let τ be an automorphism. ThenfHT ·= lim Σ (f° xk/k —/° т~к/к) exists
n-+ao k=l
a.e.forfe Ll. Similarly, if{zt: — со <t< со} is a measurable measure preserving flow, then
lim J/o Tt/tdt exists a.e., where Ιε = {t: ε ^ \t\ <Ξ 1/ε}.
ε->0 IE
Petersen [1983] gives a thorough discussion and references.
Teicher [1967] has used Wieners dominated ergodic theorem to deduce in the i.i.d. case: if
\/άμ = 0 and \\f\\p < oo, (p > 2), or || \f\p log* \f\ \\1<co,(p = 2), and if cn is a positive
00
decreasing sequence with cn = 0(n~r/2) and Σ n[r]'icn2[r]/r < oo, then sup {cn\Snf\r} is
integrable. Related results for {/-statistics appear in Ahmad [1981].
Teicher [1971] proved: If X0,XU... are i.i.d. with EXX = 0, then
и
supn^ee|Z^I7«loglog«)r/2is integrable iff E\XJ < oo, (for r>2), resp. iff
ι
X\ log | Χλ | l{|Xl| > ee}/log log | Xl | is integrable, (for r = 2). The case where r is an integer is
due to Siegmund [1969].
G. Baxter [1965] has given a subtle refinement of HopPs maximal ergodic theorem. To
state it we abreviate "for infinitely many values of £" by i.m.k, and "for finitely many
values of £" by f.m.k:
Theorem 6.17. If Τ is a positive contraction in Lufe Lu and 1 5Ξ ra 5Ξ n, then
and
J Sn/άμ ^ - J Μ^άμ
J Sm/^- J ΜΜ/άμ.
{Skf>M&fi.m.k} {Skf>M&ff.m.k}

Some general tools and principles 63
In particular
{Skf>Oi.m.k}
Continuous parameter analogues of these inequalities have been established by Berk
[1968].
§1.7 Some general tools and principles
In this section we discuss the Banach principle, the ergodic theorem for sequences
of functions, and, in the Notes, the necessity of maximal inequalities.
Л is a directed set if there is a partial order ^ (satisfying λ ^ λ and transitivity,
but not necessarily λ1^λ2^λ1 => Ax = λ2)9 and if for λ,λ'βΛ there exists A"
^ Я, λ'. E.g., Л = Z+ or Л = (Z+)d with partial order (щ) rg (и,·) iff щ rg vt for
i = 1,..., d. In the proof of von Neumann's theorem we have implicitly used:
Proposition 7.1. Let Л be a directed set, and let Τλ, (λ e Л), be a family of linear
operators mapping a Banach space X into a Banach space Ϊ). If the Τλ are uniformly
bounded, (i.e., //sup || Τλ\\ < со), andifTxx converges in norm for all χ in a dense
subset ofX, then Τλχ converges in norm for all xeX.
The proof is obvious. There also is a converse: If the Τλ are bounded and Τλχ
converges for all x, then the norms || Τλχ\\ are bounded for each fixed x, and,
by the uniform boundedness principle, the Τλ are uniformly bounded.
7^χ := Нгпд Τλχ then defines a linear operator with || 7^ || ^ sup || Τλ||, (theorem
of Banach Steinhaus).
One of the most useful tools in the study of pointwise ergodic theorems is
Banach'sprinciple, which, similarly, allows to deduce almost sure convergence
for all χ e X from almost sure convergence for all χ in a dense subset. In this
principle, Ϊ) is a space of equivalence classes of functions and the uniform
boundedness of the Τλ is replaced by a pointwise condition stated below.
Let (Ω, si, μ) be a finite measure space and Ϊ) a linear space of equivalence
classes of real or complex valued measurable functions on Ω. A map Τ: Χ -* Ϊ) is
called continuous in measure if, for any sequence xn e X converging in norm to
some x, and for any ε > 0, μ(\Τχη — Tx\ > ε) -> 0. For a family 3~ — {Τλ, λ e Λ}
we employ the notation
М*~х = ьир{\Тлх\} Mfx = sup{\TQx\:QUX}.
ХеЛ
Л is called locally finite if {ρ e Λ: ρ ^ λ} is finite for all λ e Л. Clearly, if Л is
countable and locally finite, the a.e.-convergence of Τλχ implies the finiteness a.e.
ofM^x.

64 Measure preserving and null preserving point mappings
We shall only need the easy half (a) of the following theorem in the sequel:
Theorem 7.2 (Banach principle), (a) Let Λ be a countable directed set and 2Γ
= {Τλ, λβ Λ} a family of linear operators from X into Ϊ). If there exists a positive
decreasing function С (a) with lima_00 С (a) = 0 and
(7.1) μ{Μ*χ > a ||x||) ^ C(a) Va > 0, χ e X,
then the Τλ are continuous in measure, and the set of elements χ e Л for which Τλχ
converges a.e. is closed.
(b) Assume Λ countable and locally finite. If the Τλ are continuous in measure
andM^ χ is finite a.e. for all χ e X, then there is a positive decreasing function С (a),
(a > 0), tending to 0 as a -* oo, and satisfying (7.1).
Proof, (a) The continuity in measure of the Τλ follows from μ (| Τλχη — Τλχ\ > ε)
й μ{Μ^{χη - χ) > ε) S С(е\\хя - χ\\~χ) -* 0, as ||jc„ - jc|| -* 0. Now let ζ be a
limit of elements zn of X for which Χ\\ΆλΤλζη exists a.e.. Put
Δ{ω,χ)·.= Υ\™$ννλίρβΛ\Τλχ(ω)-Τρχ(ω)\ We have \A(; z) - A(-, zn)\
^ Δ(·9ζ — zn) rg Μ^(ζ — zn). Hence, for any ε > 0, μ {ω: | Α (ω, ζ)\ > ε}
й μ(Μ*"(ζ - ζη)> ε) й C(s\\z - ζη\\~ι) -* 0. Hence, Λ(·,ζ) = 0, and \imxTxz
exists a.e..
(b) For any χ e X and ε > 0 the finiteness of Μ^ χ and of μ implies the
existence of an η with μ{Μ^χ > ή) rg ε. Thus X is the union of the sets Bn
= {χ: μ{Μ^χ > η) rg ε}. For each n, Bn is the intersection of the sets Βηλ
= {χ: μ(Μ[χ > η) ^ ε}. Mf is continuous in measure because Λ is locally finite.
Hence, each of the sets Bnд, and therefore also each Bn is closed. By the Baire
category thorem at least one of the sets Bn must have a non empty interior, and
thus must contain a closed sphere. If x0 is the center, and r > 0 the radius of this
sphere, we have
μ(Μ^χ> ή) :g ε for all χ with ||x — x0|| :g r.
For this we may write
μ{ω: M^(x0 + rz) (ω) > η} S ε for all ζ with ||z|| ^ 1.
Using
M^zu M^(x0 + rz)/r + M^x0/r
we obtain, for ||z|| rg 1, the estimate
μ (m^z ^ — J й μ(Μ^(χ0 + rz) > η) + μ(Μ^χ0 > л) ^ 2ε.
Thus, for any α ^ 2w/r, we have μ(Μ^ζ > ос) й 2ε. Hence,
C(a) := sup {μ(Μ^ζ > a): ||z|| ?g 1} tends to 0 as a -^ oo. To prove (7.1), we may
assume χ Φ 0 and pass to ζ = лг/||лг||. □

Some general tools and principles 65
In applications the operators Τλ usually are bounded linear operators in a space
Lp, (1 S Ρ < °o). Such operators clearly are continuous in measure. The measure
μ may then also be σ-finite, because the theorem can then be applied with X
= LP{Q) and Ϊ) = LP(A), where A is a subset of Ω having finite measure.
As an example of an application we sketch the proof of an extension of
Birkhoff s theorem proved by Hopf [1954] (for 74 = 1) and Dunford-Schwartz
[1956]. When Τ is the composition with an endomorphism we obtain an
alternative simple proof of BirkhofFs theorem:
Theorem 7.3. If Τ is a positive Lx — L ^-contraction in Lx of a finite measure
space, the averages Anf converge a.e. for fe Lv
Proof 74s also a contraction in L2. This is clear when 74s the composition with τ,
and it follows from the Riesz convexity theorem or from the subsequent lemma in
the general case. By von Neumann's theorem the set of functions /= g + (Th
— h) with Tg = g e L2 and h e L2 is dense in L2. We may assume he L^ replacing
h by (— Κ) ν h а К for large enough K. It follows that the set of such / with
bounded h is also dense in Lt. For these/we clearly have lim Anf= g a.e.. As
Lemma 6.1 implies (7.1) with C(a) = 1/a, the proof is complete, a
Later, we shall encounter various generalizations of this theorem.
The following lemma was shown to me by Μ. Lin. It provides a simple proof of
the fact that positive Lx — L^ contractions are also contractions in Lp for
1 <p<oo:
Lemma 7.4. If Τ is a positive linear operator in a space of equivalence classes of
measurable functions, 1 <p < oo, and p~x + q~x = 1, then
(7.2) \Т№\й[Т(\Лр)У1р[Т(\8\'>)У"> a.e..
In particular, if Τ contracts the Lm-norm, \ Tf\p iS T{\f\p) a.e..
Proof. It is known from the proof of Holder's inequality (see Hewitt-Stromberg
[1969]) that
abUP^a' + q^b9, (a,b^0).
We may assume/, g ^ 0. For any c, d> 0 we obtain
cd ~ cpp dqq '
and hence
(7.3) Т№)1са<,Т{П1срр + Т(я«)1аЦ a.e..
Fix representatives in the equivalence classes T(fg), T{fp), T(gq). There

66 Measure preserving and null preserving point mappings
is a fixed nullset N such that (7.3) holds for all rational c,d>0 in 7VC, and
- by continuity - for all c, d> 0 in Nc. Now set с = c{co) = (T(fp)) (ω)1/ρ
and d=d(co) = (T(gq)(co)1/q, and use i/p-i/q = i9 to obtain (7.2) on
{ω e 7VC: с (ω) > 0 and </(ω) > 0}. (7.2) follows on {ω: с (ω) = 0} from /g
^ lim„ л/р, and on {ω: ά{ω) = 0} from/g ^ limmm^. α
We now discuss the ergodic theorem for sequences of functions. Maker [1940]
proved a variant of
Theorem 7.5. Let τ be an endomorphism ο/(Ω, «я/, μ) and (fni) a family of
measurable functions such that supijfl \fnti\ is integrable andfni -* fa.e., (n, i -> oo). Then
' n-l
the limit of the sequence hn = n~l X fni ° τ1 exists а. е., and is equal to the limit of
A„(x)f.
Proof We may assume/= 0. Put gk = sup {\fni\: n, i^k}.
As the/mi with i ^ к and m rg к do not matter for the limit behaviour of A„, we
have lim sup„ \hn\ ^ lim„yln(T)gk for each A:. The conclusion now follows from
\\im An{x)gku JSfc^O. α
Sucheston [1983] has given a similar result for operators in certain Orlicz spaces.
To avoid various definitions we assume that the spaces L, LI below are spaces Lp,
(1 rg ρ < oo), or L logmL of a finite measure space.
Theorem 7.6. Le/ Abe a countable directed set, and let La L! be two spaces as
above. Let {Τλ, λβ Λ} be a family of positive (and hence bounded) linear operators
Τλ: L -> Z/. Assume that, for eachfe L, T^f-— limA Txf exists a.e. and is in ΖΛ Iffn
is a sequence in L with sup \fn\e L andfn ->/ a.e., then
lim ГА/В = Γβ/ а.е..
ХеЛ
пеЫ
Proof The sequence gn -·= supk^ w |/k —f\ is decreasing with limit 0. For each η we
have
lim sup | TJk - T„f\ й Ит sup \Tx(fk -f)\ + lim sup |TJ- T„f\
X,k A,k A,k
^ lim sup 7^ = 7^„.
A
Now gn i 0 implies 7^ g„ J,0. α
Notes
The idea of the Banach principle was conceived by Banach [1926]; see also Saks [1927],
and Mazur-Orlicz [1933]. Its importance for ergodic theory was realized by Yosida

Some general tools and principles 67
[1940]. We have partly followed Garsia [1970], whose book gives a splendid introduction
to the Banach principle and to several related results.
Yosida [1940a], [1974] has a lattice theoretic version of the first part of theorem 7.2,
further generalized by von Weizsacker [1974]. A special case of his results is: Assume
{Τλ, λ e A} is a set of continuous maps of a Banach space £ into a Banach lattice 2)
in which order convergence implies norm convergence. (E.g. 2) = Lp, (1 5Ξ ρ < oo)).
Assume Τλ(αχ) = ocTxx, (ос ^ 0), and Τλ(χ + xf) ^ Τλχ + Τλχ\ (χ, χ' e X). If the maps
χ -* lim sup Τλχ are well defined on X, they are continuous.
The papers of Klimko and Sucheston [1969], Klimko [1969], Mesiar [1984], and
Burke [1965] are references related to the ergodic theorem for sequences of functions.
The necessity of maximal inequalities. The maximal inequality (6.1) says that (7.1) holds
even with C(a) = a-1 for the operators Tn = An(T), when Γ is a positive Lx — L^-
contraction. This is much more than the statement C(a) -» 0, (a -> oo). Similarly, (6.5)
implies that the same family 9~ = {Tn} satisfies an estimate
Hence, the family 6Γ of operators in Lp, (1 <p<co), fulfills (7.1) with
C(a) = (p/(p — l))pa~p, and again we have a quantitative result.
We now report briefly on some results about the necessity of such stronger estimates for
rather general sequences of operators Tn. Apart from their intrinsic interest such results
can be useful tools for the construction of counterexamples. For specific sequences Tn of
linear operators in Lp, (1 ^ ρ < oo), it is frequently much simpler to show that there is no
constant К > 0 with
(7.4) μ{Μ£ f>k\\f\\^uKlk*, VfeLp
than to exhibit an/e Lp for which the sequence 7^/diverges on a set of positive measure.
The first results on inequalities of the form (7.4) as necessary conditions for a.e.-
convergence arose in Fourier analysis. Kolmogorov [1925] studied the operators
Tnf(co)= f f(a> + t)/tdt
|i|>l/n
for integrable functions vanishing outside a finite interval and deduced an inequality (7.4)
with ρ = 1. Calderon proved that the a. e.-convergence of Tnf for all square integrable/on
the unit circle is equivalent to (7.4) with ρ = 2, when Tnf is the и-th partial sum of the
Fourier series for/. E.M.Stein [1961] generalized this considerably. He considered a
compact group G and a homogeneous space Ω of G, i.e., there are maps xg\ Ω -> Q,(ge G),
with xgh = xgxh and each orbit {τ9ω: g e G} is all of Ω. There is a measure μ in Ω invariant
under all "translations" тд. (E.g. Ω = G, μ = Haar measure). In this setting he proved that
for any ρ with 1 <Ξ ρ ig 2 and any sequence of bounded linear operators Tn in Ζρ(μ) which
commute with all translations ((Tnf) °τ9=Γπ (/° τ9)) the a.e.-finiteness of all M^/implies
the existence of а К < oo for which (7.4) holds.
As many problems in ergodic theory and in probability theory do not fit into this
setting, Burkholder [1964], Sawyer [1966], and Garsia [1970] introduced several other
conditions allowing similar conclusions. The basic idea is as follows: If there is no constant
K<co with (7.4), then there are functions/" for which M^fn is very large on small sets. If
we take many independent copies of each/" or "sufficiently independent" translates they
will have this bad behaviour on other small sets and the union of these sets will have a
fairly large measure. But if/is a convex combination of the/" then M^/will be large on
this union.
W"

68 Measure preserving and null preserving point mappings
Let us make this precise in the situation discussed by Burkholder: Let D be the set of
sequences X = (Xu X2,...) of non negative random variables on a probability space
(Ω, «я/, μ). Χ ~ Υ means that X and Υ = (Yl, Y2,...) have the same joint distribution. A
family CcDis called stochastically convex if for each sequence X" = (X\, X^,...), (n
^ 1), in D there exists a sequence Yn in D with Yn ~ X" such that Y1, Y2,... are stochasti-
00
cally independent and, for each al9 a2,... ^ 0 with Σ я„ = 1, there is a Z = (Zl5 Z2,...)
fc = l
00 00
inCwithZ~(S л„*7, Σ лиГ2",...). ForAreCputAr* = sup{JTi:l ^/<oo}. For/?
π = 1 η = ί
= 1 Burkholder proved:
Theorem 7.7. Suppose that С и stochastically convex and that μ {X* < oo } > 0 holds for all
Xe С Then there exists a K< oo .уисй f/ш* μ{Χ* > α} 5Ξ ΛΓα-1 holds for all α > Ο a«d л//
JeC.
Proo/. If there is no such К and we put
Af(a) = sup{a/i{X* >a}:IeC},
then lim sup Μ (a) = oo. It is not hard to see that this implies the existence of sequences
a-*· oo
oo oo
(an) and (απ) satisfying an > Ο, Σ an = 1, ctn -> oo, and Σ (ajκη) Μ(ajan) = oo. Let
JTe С satisfy "=1 n=1
μ((*-)* > απ/απ) > (Λπ/απ)Μ(απ/απ) - 2"",
(« = 1,2,...). Take 7" as in the definition of stochastic convexity. Then μ ((Yn)* > ocn/an)
00
= μ{{ΧηΥ > αΒ/αΒ) yields Σ Κ(Υη)* > απ/Ό = oo. As the processes Yn are indepen-
n = l
dent, the Borel-Cantelli lemma shows that (Yn)* (ω) > ocn/an almost surely holds for
infinitely many n. Hence lim sup„^ю an (Yn)* (ω) = oo a.e.. Again by stochastic convexity there
00 00
exists ZeC with Ζ~ ( Σ αη*7> Σ an Y2n,...). As all terms are non negative
n=l n=l
00
sup Σ an ί7 ^ sup sup я„ У^п = sup я„(Гп)* = oo a.e..
i n=l in η
We obtain Z* = oo a.e., a contradiction to the assumption that μ(Χ* < oo) > 0 holds
for all Xe С D
There is a similar theorem for 0 < ρ < oo. Burkholder gave applications to ergodic theory,
probability theory, and orthogonal series. E.g. the family of non negative stationary
processes on the unit interval Ω and the family of non negative submartingales with
bounded integrals on the same Ω are stochastically convex. If С is stochastically convex
and С is the family of processes X = {Xu X2,...) for which there exists an Xe С with Xn
η
= η'1 Σ %h tnen ^ is easy to see that С is stochastically convex.
Garsia [1970] worked in an ergodic theoretic setting with rather weak assumptions. We
refer the reader to his book for the following continuity principle:
Theorem 7.8. Let ΖΓ = {Tn} be a sequence of linear operators in Lp, (1 <Ξ ρ <Ξ 2), of a finite
measure space (Ω, «я/, μ), which are continuous in measure. Let $ be a family of endomor-

Some general tools and principles 69
phisms τ with
(7.5) (M7)°^MJ(/n), V/eLp.
Suppose that for each A, Be я/ and each α > 1 there exists axe $ with μ(Αητ~1 Β)μ(Ω)
5Ξ ос μ (Α) μ (Β). Then the condition
М^/< oo a.e. V/e Lp
is equivalent to the existence of а К < oo vwY/z (7.4)
The condition (7.5) is satisfied if (Tnf) ° τ = Γπ(/° τ) holds for all/e Lp, τ e <f and Tn.
Sawyer [1966] has proved the continuity principle for 1 <Ξ ρ < oo, but then the operators
Tn must also be positive.
As we mentioned before, the Banach principle, the continuity principle, and similar results
can be powerful aids for the construction of counterexamples. In this way, del Junco and
Rosenblatt [1979] have obtained a unified approach to many negative results. Their paper
contains also some interesting variations of the main theme. E.g. in some instances where
theorem 7.8 allows to infer that there is an/e Ll with lim sup Tnf= oo they can obtain an
indicator function/= 1A with lim sup Tnf= 1 and lim inf Tnf= 0.

Chapter 2: Mean ergodic theory
The mean ergodic theorem deals with averages taken along the orbits of elements
of a topological vector space under semigroups of operators. In the Banach space
case we also deal with uniform convergence of the averages of operators. It can be
obtained for quasi-compact operators. Then we discuss weak mixing and
multiple recurrence. The theorem of Jacobs-Deleeuw-Glicksberg extends the spectral
mixing theorem to semigroups in Banach spaces.
§2.1 The mean ergodic theorem
We are now concerned with questions of weak and strong convergence of aver-
n-l
ages n~l Σ T*x, where Γ is a continuous linear operator in a Banach space X
i = 0
with norm || · ||. We then proceed to the study of more general semigroups of
operators in more general spaces.
1. Operators in a Banach space. Let us fix some notation: If A is a subset of a
m m
linear space, co^l or со (A) is the convex hull { Σ a^: xte A, 0 rg ah £ at
i=l i=l
= 1, m e N} of A. linA denotes the linear span of A9 the smallest linear space
containing A. We write A + В for {x + y: xe A, ye B}; if A and В are linear
spaces and А ел В = {0}, we say that A + В is a direct sum and write A © B. This
means that the representation χ + у is unique.
Recall that, for a space ϊ)2 of linear functionals on a space SQU the σ(ϊ)1? ϊ)2)"
topology is the coarsest topology on ϊ) χ with respect to which all functionals in
ϊ)2 are continuous. The weak topology is the σ(Χ, £*)-topology, where X* is the
space of continuous functionals on X. The w*-topology is the σ(£*? 3E)-topology
on X*. lim denotes the limit in the strong(= norm) topology, w-lim in the weak
topology and w*-lim in the w*-topology. cl A is the closure of A a X, and w-cl A
the closure in the weak topology. As usual, со A denotes cl со A. Note that this
agrees with w-co^l = w-cl со A, (Dunford-Schwartz [1958: 422]).
<x, A> is the value of the functional h e X* in χ e X. Recall that if (X) is the
space of bounded linear operators Τ in X, and that T° = / is the identity
operator. The adjoint or dual operator of Γ is the operator Г*: Ж* -* £* with
<7x, h} = <x, Γ*h} for xeX.he X*.
Tis called power bounded if the norms of the powers Tk, (к ^ 0), are uniformly

72 Mean ergodic theory
bounded (supfc || Tk\\ < oo), and Cesaro bounded if the norms of the averages An
n-l
= An(T) = n~l Σ Γ1 are uniformly bounded, a clearly weaker condition.
i = 0
Because of the identity
(l.i) ,,-ιγ»-1^,,- —4,-1
Π
the condition
(1.2) Шп/Г1Г,-1* = 0
is necessary for the convergence of Anx. It is satisfied for all χ when Τ is power
bounded.
Theorem 1.1 (Mean ergodic theorem). Let Τ be a Cesaro bounded linear operator
in a Banach space X. For any xeX satisfying (1.2), and any yeX the following
assertions are equivalent:
(i) Ту — у andjeco{x, Tx, T2x,...},
(ii) у = \\mnAnx,
(Hi) у = w-lim^x,
(iv) у is a weak cluster point of the sequence (Anx).
Proof The implications (ii) => (iii) => (iv) are trivial, (i) => (ii): Set Μ·= sup„ \\An\\.
For ε > 0, (i) implies the existence of an operator S в со {Г°, Г1, Г2,...} with
|| у — Sx\\ < ε. For any к and any η we have
k-l k-l
AnTkx-Anx = n~l Σ Τη+ίχ-η-χ Σ Τιχ.
i = 0 i = О
Using (1.2) and (ι + ή)/η -* 1 we see that ||у!пГкх — ^4nx|| < ε holds for large
enough n. As 5 is a convex combination of finitely many Tk and {z e Χ: || ζ || < ε}
is convex, we may find an N such that 11 An Sx — An χ \ \ < ε holds for η ^ N. As у is
a fixed point of Τ we have ^4W>> = у for all л. For η ^ TV we therefore obtain
ΙΙ^-Λ^ΙΙ^ΙΙΛί^-^ΙΙ + ΙΙΛ^-Λ^ΙΙ^Μβ + β.
(iv) => (i): We now need Mazur's theorem which says that any closed convex
subset of Ж is also weakly closed. The weak cluster point у of the sequence Anx of
convex combinations of x, Tx,... therefore belongs to со {χ, Τ χ,...}.
To prove у—Ту take any he X* and ε > 0. By the argument used above we
know TAnx — Anx -> 0. For large enough η
\(TAnx-Anx,hy\<e.
As у is a weak cluster point of (Anx) there exist arbitrarily large values of η with
\<y,h>-<AHx,h>\<e and \<y, Γ*Α> - (Anx9 T*h}\ < β.

The mean ergodic theorem 73
Combining these estimates we arrive at | <>>, h} — <7y, h} \ < 3ε. But ε and h
had been arbitrary, α
The present formulation of the mean ergodic theorem is a special case of results
of Eberlein [1949], but the main assertions emerged already with the work of
F. Riesz [1938] for X = Lp, and independently with the work of K. Yosida [1938]
and S. Kakutani [1938] for general Banach spaces; see also Yosida-Kakutani
[1941]. The following important special case was proved by E. Lorch [1939]
independently:
Theorem 1.2. If Τ is a power bounded linear operator in a reflexive Banach space
X, the averages Anx converge in norm to a T-invariant limit for all xeX.
Proof The power-boundedness implies (1.2), and each norm bounded set in a
reflexive Banach space has a weakly compact closure. By the Eberlein-Smulian
theorem, it is then weakly sequentially compact, α
Note that the spaces Lp, (1 <p < oo), are reflexive. Theorem 1.1 also contains a
mean ergodic theorem in Lx: If Τ is a contraction in Lx and there exists a strictly
positive integrable function ρ such that \Tf\ Sp holds for all/with \f\ rg/?,
then Ang converges in Z^-norm for all ge Lx. (Proof For any ε > 0 there is а с
and a splitting g = gc + g'c with \gc\ ^ cp and \\£€\άμ<ε. This implies
J (| Ang | — cp)+ άμ < ε for all n. Thus, the sequence (Ang) is uniformly integrable
(Def. 3.3.13), and therefore weakly sequentially compact; see Dunford-Schwartz
[1958].) The application of theorem 1.1 to C(K) is treated in § 5.1.
Our next aim is a splitting theorem for
%me = %me(T) '-= {x 6 X: Иш A„X exists} .
Clearly, if Τ is Cesaro bounded, Xme is a closed linear subspace of Χ. Τ is called
mean ergodic if X = Xme. We shall use the notation
F= F(T) := {xeX:Tx = x}, N-.= {x -Tx:xeX} = (I- T)X
F* = F*(T)i= {h e **: Γ*h = A}, TV* := (/- T*)X*.
Most of the next theorem is due to Yosida [1938]:
Theorem 1.3. Let Τ be Cesaro bounded, and assume that (1.2) holds for all xeX.
Then Xme = F 0 cl N. The operator Ρ assigning to xe Xme the limit Ρ χ ··= lim Anx
is the projection ofXme onto F We have Ρ = Ρ2 = Τ Ρ = Ρ Τ For any zeX the
assertions
(i) lim Anz = 0,
(ii) <z, A> = 0 for all he i^,
(Hi) ζ e cl TV
are equivalent.

74 Mean ergodic theory
Proof. Clearly Fand clTV are linear subspaces. Let us verify Fn clN = {0}: For
ε > 0 and ζ e FnclN there exists и with ||z — (u — 7w)|| < ε. Hence
\\An(z -{u- Tu))|| < Με. Using ,4nz = ζ and ,4w(w - Tu) -+ 0, we find
||ζ|| < Με + ε. A similar argument shows F^nclN^ = {0}.
For xe£me, theorem 1.1 implies PxeF. Thus z = x — Ρ χ satisfies (i). (i)
=> (ii): heF# implies h = A*h for all η and hence <z, h} = <z, ^4*A>
= <^nx, A> -* 0.
(ii) => (iii): If ζ does not belong to cl N there exists, by Hahn-Banach, an h e £*
with <z, й> ф 0 and with <>>, A> = 0 for all >> e clN. In particular, <w — 7w, A>
= 0 for и e X. Hence <w, h — Г* h} = 0 for all w. This implies A e i^, a
contradiction to <z, A> Φ 0.
We have proved Xme a F © cl N. As F cz Xme is trivial, the opposite inclusion
will follow from (iii) => (i): For any ueXwe have An(u — Tu) = n~x(u— Tnu)
and this tends to 0. Thus, all ζ e TV satisfy ylnz -* 0. But the set of ζ with this
property is closed because Τ is Cesaro bounded. As Ρ is the projection of
F © cl N on Fand the elements of Fare fixed under Γ the identities Ρ = Ρ2 = TP
are clear. Ρ = PT follows from An{x — Tx) -► 0. α
Recall that a linear operator Q is called projection if Q = Q2. A projection with
Q = QT = TQ will be called T-absorbing. Thus, Ρ is a Γ-absorbing projection.
Occasionally, the following criterion of Sine [1970] for mean ergodicity is
useful:
Theorem 1.4. Let Τ be Cesaro bounded and assume (1.2) for all χ. Τ is mean
ergodic iffF separates F^.
Proof. First assume convergence. If hl9 h2 e F^ are distinct there exists апхеЗЕ
with <x, hxy φ <x, A2>. Now Ρ χ belongs to Fand separates the ht because of A,-
= P*ht and <Px, ht} = <x, Р*йг>. Now assume that i7separates F^.IfF® clN
is not all of X there exists a non zero heX* with < >>, A> = 0 for all у e F © cl N.
In particular, <x — Tx, h} = 0 for all χ shows he F^. As F separates h and 0,
there exists у e F with <>>, h} Φ 0, a contradiction, α
As the case of reflexive X suffices for many applications we end this subsection
with a sketch of Lorch's elegant argument. Assume Γ power bounded in a
reflexive space X:
For Ε с X set Ε1 = {A e X*: <x, A> = 0 Vx e E}. Using Hahn-Banach and X
= X** one easily checks (TV1)1 = cl N. From h e Ν1 ο <x - 7x, A> = 0 Vx о
<x, A> = <Гх, A> = <x, T*h} Vx <^> Г*А = h we see TV1 = /ς, and hence /ς1
= clTV. By duality F1 = clTV*.
To establish X = F © cl TV it remains to make sure that any h e (F © cl TV)1
vanishes. But this now follows from he F1 = clN# and he N1 = F^.

The mean ergodic theorem 75
2.5^-ergodic nets in topological vector spaces. We shall now derive similar results
as above for more general semigroups, and we allow that they act in a locally
convex topological vector space.
We recall just a few concepts: A real valued function/? on a real or complex vector
space X is called seminorm if it is subadditive ana positively homogeneous, i.e., it
satisfies ρ (χ + у) йр(х) -\-р(у) for all χ, у е X,a,ndp(<xx) = |a|/?(x)forallxe Χ
and all scalars a. A family {py: уеГ} of seminorms satisfies the axiom of
separation if, for any χΦΟ, there exists some у with/?y(x) Φ 0. Fis a neighbourhood of
χ if there are finitely many yt and st such that V contains all у mthpy. (y — x) < 8t.
With this topology, X is a topological vector space, i.e., the mappings (x, y) -> χ
+ y9 (a, x) -* ax are continuous. X is called a locally convex topological vector
space (l.c.t.v.s) if there exists a family {py: уеГ} of seminorms which satisfies the
axiom of separation and generates the topology in X. For more detail and for the
functional analytic tools needed in the sequel we refer to Yosida [1974] and
Schaefer [1966].
Let Jf{y) be the family of neighbourhoods of y. A family {Tk} of linear
operators in X is called equi-continuous if, for any V e ^T(O), there exists some
We JT{Q) such that Tk Wen Fholds for all k. E.g., a set if of linear operators Τ
in a Banach space is equi-continuous iff it is bounded, i.e., there exists an Μ < oo
with || ГЦ й Μ for all Те У.
We say that a semigroup if of continuous linear operators Tin X admits a right
if-ergodic net {Αλ: λ e A} if A is a directed set and
(El) each Αλ is a linear operator in X,
(E2) for each xeX and all λ, Αλχ e coif χ,
(E3) the A'xs are equi-continuous, and
(E4r) for every χ e X and Τe if, limA (Αλ Tx - Αλχ) = 0.
We say that the net is left if-ergodic if (E4r) is replaced by
(E41) for every χe X and Те if, Χναχλ{ΤΑλχ - Αλχ) = 0.
It is called if-ergodic if it is right and left 5^-ergodic. If lim in condition (E4r) is
replaced by w-lim the net is called weakly right if-ergodic, etc.
Perhaps the most important example is a multiparameter semigroup. Let
Ti9..., Td be commuting power bounded operators in a Banach space. Put Λ
= Nd. if is the semigroup generated by TX,T2,..., Td. For λ = (λί9..., λά) we
may take
d Ai-l Ad-1
^я = (ПЯ,Г1 Σ .·· Σ 7?'...#-.
ν=1 £ι =0 id = 0
We may also take averages over convex sets like spheres instead of rectangles.
It is less obvious that 5^-ergodic nets exist for arbitrary Abelian equicontin-
uous semigroups if\ We may then take A — со if and Αλ = λ. A partial order in
A is defined by Τ ^ S iff there exists an R e A with Τ = RS. It follows from RS
= SR that Л is a directed set. The net clearly has the properties (E1)-(E3). (E4):

76 Mean ergodic theory
For Αλ ^ An(T), i.e., for all large enough λ, there exists an Re со5^ with Αλ
= RAn{T), and then Αλ-ΑλΤ= Д(и_1(Г0 - Tn)\ This clearly yields (E4r),
and by commutativity (E41).
In most applications the question of existence of a net does not arise because
the interest lies in convergence for a particular given net.
Theorem 1.5 (Eberlein [1949]). If*£f is a semigroup of continuous linear operators
in a l.c.t.v.s. £, and admits апУ-ergodic net {Αλ: λ e Л}, then, for any x,yeX, the
following conditions are equivalent:
(i) Ту = у for all Те У and у е со 5^ χ,
(ii) у = ШхАхх(=*Рх),
(Hi) у = w-limxAxx,
(iv) у is a weak cluster point of {Αλχ: λ e Λ}.
The proof of theorem 1.1 has been written in such a way that it carries over to the
present more general situation with minor modifications (e.g. ε-neighbourhoods
are replaced by topological neighbourhoods).
It seems to be unknown if the equivalence of (i)-(iv) persists if one of the
conditions (E4r), (E41) is dropped. (E4r) was needed for (i) => (ii), and (E41) for
(iv) => (i). If we assume only that the net in theorem 1.6 is weakly 5^-ergodic, the
conditions (i), (iii), (iv) remain equivalent.
We mention two applications of Eberlein's theorem:
(1) Existence of the mean for continuous almost periodic functions. Let X be the
Banach space of bounded continuous functions x: R -* R with the usual norm
|| χ || = sup | x(t)\. We consider the group У = {Tt9 te R} of translations (Ttx) (·)
= x(· + t)9 and the averages
Ллх() = Я-1}т;х()Л.
о
Taking von Neumann's definition of almost periodicity, χ is almost periodic \i£fx
is conditionally compact. Then со Zfx is compact. It is rather straightforward to
check that {Αλ: λ > 0} is an 5^-ergodic net. As Αλχ belongs to со £fx for all λ > 0,
the equivalence of (iv) and (ii) in Eberlein's theorem yields \\Αλχ — Px\\ -> 0,
(Я -* oo). Because of TtP = P, (te R), the function Px must be constant.
(2) Fejer's theorem. As the equicontinuity has only been required for the
operators Αλ9 and not for Sf9 we may view Fejer's theorem as an ergodic theorem:
This time X shall be the space of continuous real valued functions on R of
period 2π. We again use ||x|| = sup |x(/)|. Let Snx be the n-th partial sum of the
η
Fourier expansion of x. Set Dn(t) = (1/2) + Σ cos ν/. Then
v=l
π
S„x() = l/π j x(· + t)D„{t)dt.

The mean ergodic theorem 77
The semigroup if shall consist of the identity / and the transformations Un = I
— Sn. The semigroup property follows from UnUm = Uk with к = max (га, л).
π
As the norm of Sn is the Lebesgue constant Ln = π"1 J |Z>n(w)|dw and L„
— π
= (4/π2) log л, (see Zygmund [1935: 67]), the semigroup if is not equicontinu-
ous.
Nowput^A = (« + l)_1 Σ Uy9{keA = Ν). It is not hard to check (El), (E2),
v = 0
and (E4). Let Kn be the Fejer kernel:
Kn(t) = (n + iy1 Σ Д(0.
v = 0
A simple computation shows
π
Αλχ{·) = χ(·)-π_1 J χ(· + /)Α:Λ(/)Λ.
— π
π
As ΑΓη has the properties Kn ^ 0, and π-1 J Kn(i)dt = 1, (see e.g. Zygmund
— π
[1935:88]) we clearly have \\Αλ\\ ^ 2 for all λ. Thus, (E3) holds, too. The uniform
continuity of χ and the properties of Κλ imply the equicontinuity of the functions
Αλχ9 (λ e Л). By Arzela-Ascoli {Αλχ: λ е Л} is conditionally compact. Eberlein's
theorem therefore yields the convergence of Αλχ to an element у e X with Uky
= у for all k. As у = 0 is the only element of £ fixed under if we obtain
n-l
Мл*11 -* 0, for all χ e X. This is equivalent to \\ri~1 Σ Syx — x\\ -> 0, the
assertion of Fejer's theorem. v = 0
3. The splitting theorem for semigroups. Next we shall derive a semigroup
generalization of theorems 1.3 and 1.4. We assume that X is a complete l.c.t.v.s.
[Recall that a sequence (χρ) indexed by the elements of a directed set is called a
Cauchy net if, for any V e Jf (0), xQ — χσ belongs to Ffor all large enough ρ, σ. Χ
is complete if each Cauchy net converges to an element of X~\. Set
F= {xeX:Tx = x for all Те if},
N = lin {x -Tx.xe X, Те if),
/ς = {A e X*: T*h = h for all Те if}.
The elements of Fare the fixed points under if, and the elements of F^ are the
fixed points under if* = {Г*: Те if).
We assume only the existence of a weakly right 5^-ergodic net (Αλ: λ e Λ). Put
D = {x e X: w-\\mxAxx exists},
Px = \ν-ΙϊτΆλΑλχ, (xe D),
Z>(0) = {xeD:Px = 0},
D(F) = {xeD:PxeF}.

78 Mean ergodic theory
Proposition 1.6. If Л is a complete l.c.t.v.s. and (Αλ: λ e Λ) a weakly right Sf-
ergodic net for the semigroup У, then
(a) D is a closed linear subspace ofX, and TD a D holds for all Те if,
(b) PD a D, and Ρ is linear and continuous on D,
(c) PT=P on Dfor all Те У.
Proof It is clear that D is a linear subspace. We show that it is complete and hence
closed. Let (χρ) be a Cauchy net in D, and χ its limit. The equicontinuity of the
Ax9s implies the continuity of P. Therefore (Ρχρ) is a Cauchy net. Let у be its
limit.
If [/is an arbitrary weak convex neighbourhood of 0, there exists ρ0 such that
Pxeo-ye(i/3)U and Αλ{χ - хв0)е (1/3) C/
holds for all λ. Because of Ρχρο — w-\\mAxxQo, there exists a λ0 with
A,xeo-Pxeoe(l/3)U foralU^V
But then
Λλχ - У = \Αλ{χ - *J] + ΙΑλχβ0 - Ρχρο] + 1Рхв0 - y~]
belongs to U for λ ^ λ0. As U was arbitrarily small we see that у = w-limAxx
exists and (a) is proved.
By (a) the set со Ух is contained in D for χ e D. (E2) therefore yields ΑλΌ с D
for all A, and (b) follows because D is weakly closed, (c) is a consequence of the
weak right ^-ergodicity of the net. α
Corollary 1.7. (a) F,D{F), and D(0) are closed subspaces of D with F
= PD(F)czD(F)c=:D;
(b) TD(F) czD(F)for all ТеУ, and AXD(F) с D(F)for all λ;
(c) on D(F) we have TP = P = PT= P2 for all Те У.
Proof Obvious. (It seems to be unknown whether D = D(F) holds.)
Lemma 1.8. Z>(0) = clN.
Proof The equicontinuity of the Ax9s shows that D(0) is closed. It contains the
elements x—Tx because of w-Y\m(AxTx — Αλχ) = 0. Therefore c\N is
contained in D (0). If some x0eD (0) does not belong to cl N there exists an h e X*
with <x0? A> φ 0 and <z? h} == 0 for all ζ e cl N. As <x - 7x, A> = 0 is valid for
all χ and T, h must belong to F+. Hence (x0,h} = <x, A> holds for all χ e со^л^
and by (E2) for all χ = Αλχ0. But then (Αλχ0, h} does not tend to 0,
contradicting xoeD(0). D

The mean ergodic theorem 79
An operator Q is called if-absorbing projection ifQ — Q2 — TQ — £? Γ holds for
all Те if.
Theorem 1.9. (Koliha-Nagel-Sato). Let Л be a l.c.t.v.s. and (Αλ, Xe Λ) a weakly
right if-ergodic net.
(a) D{F) is a closed linear subspace with ifD(F) с D(F), D(F) = F® clN,
and P: D(F) -> F is an if-absorbing projection.
(b) D(F) = X iff F separates F#.
(c) If% is a closed linear subspace ofX with ifty с Ϊ), and Q an if-absorbing
projection on Ϊ) with Qx e со if χforall x, then ?) с D (F), and Q agrees with Pon^.
(d) If the net is even weakly if-ergodic, D — D{F).
(e) If the net is right if-ergodic, ИтАлх = Ρ χ for xe D(F).
Proof, (a) follows from proposition 1.6, corollary 1.7 and lemma 1.8.
(b) Assume that F separates F^. If D{F) — X fails, there exists 0 φ he £*
with <x, h} = 0 for all xeD(F). By lemma 1.8 and Z>(0) с D(F) we obtain
(y — Ту, A> = 0 for all у e X and Те if, and hence he F^. As F separates F^,
there exists some xe F with <x, A> Φ 0. This contradicts Fez D(F).
Next assume D(F) = £. We have to show that, for any 0 φ h e F#, there exists
some ze F with <z, A> φ 0. Because of Α Φ 0 there exists an xeF with
0 φ <x, h}. heF^ implies <x, A> = (x,S*h} for all Seco^. In particular,
<x, A> = <x, Afh} = <у!лх, A>, and hence <x, A> = <i>x, h}. By Ρ =TP the
element Px belongs to F, and we can take ζ = Px.
(c) Restricting everything to Ϊ) we may assume X = Ϊ) and must show £
= D(F) and Ρ — Q. Take A, x as in the previous paragraph. Then <x, A>
= (Sx,h} for all Seco^, and the assumption Qxecotfx implies <x, /*>
= <{?x, A>. By (? = Γ(2 the element Qx belongs to F. As 0 φ h e F^ was
arbitrary, F separates F+9 and X = D(F). It follows from QT = TQ = Q that
Ллб = £Мд = Q for all Я and hence PQ = QP = Q. Similarly Ρ if = P and
Qx e со ifx yield Pg = P. Hence Ρ = Q.
(d) If (ΛΑ) is ^-ergodic TPx = Px holds on Z>.
(e) Passing to x — Px we may assume xe D(0) = clN. Clearly, the set
{x: lim Αλχ = 0} is closed by the equicontinuity of the Ax's. It contains TV because
lim (Αλ у — Αλ Ту) = 0 holds for all у. α
Note that the description of D (F) as a direct sum F © cl N is independent of (у!л).
In particular, if there are two nets (Αλ), (Α'ρ), convergence to fixed points holds for
the same set of elements, and the limits are the same.
4. General semigroups. The results above depend on the existence of weakly right
5^-ergodic nets. The question of existence of such nets leads to the study of right
amenable semigroups; see §6.4. However, a different argument yields fixed
points for general semigroups:

80 Mean ergodic theory
Theorem 1.10. (Alaoglu-Birkhoff). IfSf is a semigroup consisting of contractions
in a Banach space X with uniformly convex norm, and if the norm ofX* is strictly
convex, then, for each x9 шУх contains a unique fixed point Px.
Proof Let (xn) be a sequence in C--=c6^x with ||x„|| -* a:=inf {||x||: xe C).
The uniform convexity of the norm implies that (xn) is a Cauchy sequence. Thus,
С contains a point x' of minimal norm, and x* is 5^-fixed.
Assume x" is another fixed point in С Find heX* with <x' — x", h}
= || x' — x" ||. Using the w*-compactness of со 5^* h we find a point И of minimal
norm in со 5^* h. From x\ x" e С and the fact that И is fixed under со 5^* we may
infer <*, A') = <*', h'} = <*", h'}. Hence <*' - x", Л'> = 0. As x' - x" is fixed
for со Sf, and И e со 5^*A, we have <x' — x", A'> = <x' — x", A>, and therefore
x' = x". и
Remarks. Similarly, one checks that P*h = h' in co^*A is unique, that X
= Р£®кег(Р), where ker(P) is the kernel {x: Px = 0}, and X*
= P*X* 0 ker(i>*). PX and ker(i>*), and also ker(P) and i>*£* are
orthogonal to each other; see Jacobs [1960: theorem 1.2.З.]. Alaoglu-Birkhoff have
shown by example that χ' Φ χ" may happen above if the condition that X* has a
strictly convex norm is deleted.
For the interpretation of their theorem Alaoglu and Birkhoff say that a
sequence χλ indexed by the elements Я of a set Л which is only partially ordered
(and not necessarily directed) converges to y, if for each neighborhood U of у and
each λ there exists ζ ^ λ such that χη belongs to £/for all η ^ ζ. They consider the
partial order in Л = со if defined by λ ^ ζ iff there exists ξ e Λ with ζ = ξ ° λ. (Λ
was directed in the Abelian case.) It is an exercise to show that the convergence of
xx = Axfor all χ is equivalent to the existence of a unique fixed point in со if χ for
all x.
Theorem 1.9(c) and theorem 1.10 motivate the following result which will be
used in section 9.1. Recall that the weak and strong operator topologies are defined
in if (£) by wo-lim Τλ = Γ, (so-lim Τλ = Τ), iff w-lim Τλχ = 7x, (lim Τλχ = Tx),
for all x; with λ e Λ, a directed set. The w*-operator topology is defined in if (£*)
byw*o-lim7^ = r'iffw*-lim T[h = Τ h for all h e X*. со ^denotes the closure
of со if in the strong operator topology. It agrees with the closure in the weak
operator topology (Dunford-Schwartz [1958: 477]).
Theorem 1.11 (Nagel). Let if be a bounded semigroup of linear operators in a
Banach space X. Then the following properties are equivalent:
(i) coif contains an У-absorbing projection P;
(ii) w*o-co if* contains an if*-absorbing projection P' which is w*-
continuous;
(iii) ooifxnF^tyfor all xeX, andcoif*hnF^ * Φ for all he X*;
(iv) F separates F^, and coif* hnF^^ Φ for all he X*.

The mean ergodic theorem 81
Proof. The equivalence of (i) and (ii) follows because the w*-continuous linear
operators in X* are the adjoints of the elements of if (X). We may take F = P*.
Ρ e со if is equivalent to P* e w*o-co5^*.
(i) => (iii): Ρ χ belongs to Fncoifx, and P*h to i^ η со 5^* Α.
(iii) => (iv): We argue as before. For 0 φ A e F^ there exists χ with <x, A> Φ 0.
Take x' e Fnco^x. For any Τe со 5^ we have <7x, A> = <x, Г*А> = <x, A>.
Hence <*', A> = <x, A> Φ 0.
(iv) => (ii): If A', h" dim two elements in F^ η со 5^* A, there exists xeF with
<x, A'— A"> φ 0. xei7 and A', A" ecu У7* A again implies <x, A'> = <x, A>
= <x, A">, a contradiction. Thus i^ η со 5^* A contains exactly one element,
which we denote by Ρ' h.
Put Af=sup{||7T||:7Te^}. To show F(ht + A2) = Fhx + P'A2, observe
that for any x1?..., χη, ε there exists i?' e со 5^* with | (xi9 i?'Ax — P'Ax> | < ε.
There exists A' e со Sf*R'h2 r^F^. By the uniqueness of P'h2 we have A' = Pfh2.
Hence, there exists УесоУ* with |<Х;, i>'A2 - S"i?'A2>| < ε. Now S'P'Ai
= Р'Л! yields
+ |<*„3"Д'Л2-Р'Л2>|^Ме + е.
Thus, each w*-neighborhood of P'h1 + P'h2 contains an element of
со &*(НХ + A2). Hence Fht + Fh2 = F(h1 + A2). It follows that F is linear. It
is simple to check that P* is an 5^*-absorbing projection. Now consider a net (Ал)
with w*-lim Ал = 0. As the norm of any element of со 5^* is ^ M, we have
| <x, F AA> | rg Μ| <x, AA> | -* 0. Hence, P' is w*-continuous, and (ii) is
proved, α
5. More on fixed points. An operator Γ in a vector space X is called affine if
Τ (ax + (1 - u)y) = ocTx + (1 - а) Ту holds for all x, у e X and 0 й α й 1.
Theorem 1.12 (Fixed point theorem of Markov-Kakutani). Let К be a convex
weakly compact subset of a topological vector space X. Let У be a commuting
family of continuous affine maps T.X-+X with TK cz K. Then К contains a point ρ
fixed under all ГеУ.
Proof The operators An{T) map AT into К and commute. Therefore, the sets
АП1{Т1)АП2(Т^...АПи(ТдК with nteN, TJe^, form a decreasingly filtered
family of non empty weakly compact sets, and there exists a/? in the intersection
of these sets. For any Те У and лe 141 there exists qne К with Tp—p
= TAn(T)qn- An(T)qn = n~l{Tn - I)qne n~l (К- К). As {<u,h}:ueK} is
bounded for each A e £*, we must have <Tp — p, A> = 0 for all A, and hence Tp
= p. π

82 Mean ergodic theory
Next, let us look at the fixed points of convex combinations of commuting
operators. We need the following lemma of Kakutani:
Lemma 1.13. The identity I is an extreme point in the convex set consisting of all
contractions in a Banach space X.
Proof. If / is not an extreme point there exist contractions Tx Φ T2 with
(71 + T2)/2 = I. For T= Tx - /we then have Tx = 1+ T, T2 = /- Γ and ΓΦ 0.
For any heX* put ht = T{*h. As Tt* is a contraction, ||/г£|| ^ \\h\\. If h is an
extreme point of В = {h e Ж*: ||h\\ rg 1}, the identity h = (hi + h2)/2 shows h
= hx = hl9 i.e., Г* A = 0. By Krein-Milman jB is the closed convex hull of the set
of extreme points. It follows that Г*А = 0 holds for all he B. But T* = 0
contradicts Γ Φ 0. α
Recall the notation F(T) = {x e X: Tx = x}.
Lemma 1.14 (Brunel-Falkowitz). Le/ {a/7 e «^} ^ strictly positive numbers with
Saj = 1 and let 7J &e commuting contractions in a Banach space X.lfT — Σα;7/
/Леи
F(r) = n^,(75)·
Proof. The inclusion => is obvious. To prove ^(Г) с ^(Tj) put
5 = (1 - a,·)-1 Efc+j-ajk7k· For * e ^(Г) we have 7Jx = TjTx = TTjX, and hence
7jx g F(r). Similarly, the contraction S maps F(T) into itself. But on F(T), Τ is
the identity. Because of Τ = α,- 3} + (1 — α,·) 5 the previous lemma shows that 7J is
the identity on F(T). α
The next result is essentially due to Sine [1975]:
Theorem 1.15. Convex combinations of finitely many commuting power bounded
mean ergodic operators Tl9 ...9Tn in a Banach space X are mean ergodic.
Proof Passing to the equivalent norm
|||х||| = 8ир{||^...Г^х||:А:^0},
we may assume that the T{ are contractions. We may also assume η — 2. By the
previous lemma and by theorem 1.4 it is enough to prove that Ρ{Τχ)ηΡ{Τ2)
separates F(TX*) η F(T2*). Let hl9 h2 be two distinct elements of F(TX*) η F(T2*).
As Tx is mean ergodic there exists an хеР{Тх) with <x, hx} φ <x, h2}. Put
xO0 = limAn(T2)x. As T2x = T2Txx = Τγ T2x shows Т2Р{ТХ) с F(TX\ we
have x^e FiT^nFiTj). x^ separates hx and A2 because of (х^КУ
= lim (An(T2)χ, ht} = Нт<х,Л№*)^> = <*? Af>. π

The mean ergodic theorem 83
Notes
Continuous parameter semigroups and resolvents. If Sf = {Tt, t > 0} is a semigroup with a
continuous parameter, ergodic theorems are frequently formulated for averages
t 00
A0ttx = t'1 \Tsxds or XRkx = X\e'ktTtxdt
о о
where t tends to oo and λ to 0 + 0. For the sake of simplicity let us assume that the Tt are
contractions in a Banach space, and that Sf is continuous in the strong operator topology
for t > 0. Ergodic theorems follow from theorem 1.5, see Eberlein [1976]. They can also
easily be deduced by applying theorem 1.1 to 7i and Аол х. The convergence of λΚλχ may
be obtained by a reduction to the case of averages A0tx; see § 8.2.
It is more interesting that different conditions for convergence may be described in
terms of the generator A of a semigroup. Consider a semigroup £f — {Tt: t ^ 0} of
contractions in a Banach space X, which is continuous in the strong operator topology.
Assume T0 = L The set dom(v4) of points χ for which Ax--=\imt^0+0/_1(7Jx —x) exists,
is dense in X. It is clear that F— {xeX: Ttx = χ for all t} is contained in ker(v4)
= {xe dom(A): Ax = 0}. It follows from semigroup theory that F= ker(v4); see e.g.
Davies [1980: 3]. If both Sf and the adjoint semigroup £f* are continuous in the strong
operator topology, the condition that F separates F^ may therefore be expressed as:
(c'): ker (^4) separates ker(v4*), where A* is the generator of £f*.
Hille and Phillips [1957] have proved that also each of the following two conditions is
equivalent to the norm convergence, for all x, of A0tx\
(HP1) For all χ in a weakly dense subset of X the set {λ(λΙ— А)~хх: 0 < λ £Ξ 1} is
conditionally weakly compact;
(HP2) cl (ker (A) + A dom (A)) = X.
(See also Davies [1980]).
It may be of interest that the convergence of A0ttx for all χ does not imply the
convergence of the discrete averages An{T^)x for all x. As an example let X be the space of all
Lebesgue integrable real valued functions on R with integral 0, and (Ttx) (·) = x(- + t).
Shaw [1980], Sato [1981], and Kataoka [1981] considered a sufficient condition for the
mutual implication of discrete and continuous parameter convergence.
Some authors (e.g. Cohen [1940], Hille [1945]) have dealt with theorems asserting that
convergence holds for some averaging method (e.g. ^ο,ί·*) iff it holds for some other
method (e.g. λΚλχ). Such results usually follow from Eberlein's theorem because the first
condition there does not involve the Ax's. However, it may happen that the boundedness
conditions are distinct. Call Τ Abel bounded if sup || AW^|| is finite, where
λ>0
Wkx= Σ (Я + 1)-(п+1)Гих.
л = 0
Cesaro bounded operators are Abel bounded, and the converse holds for positive Tin a
Banach lattice; see Emilion [1984].
A family (Vx, λ > 0) of elements of $£ (X) is said to satisfy the resolvent equation Ίϊνλ—νμ
= (μ — λ) Vx νμ, (λ, μ > 0). Then (λνλ,λ> 0), or sometimes (Vx), is calledpseudo resolvent.
The most important examples are the resolvents (λΙΙλ) of semigroups. Yosida [1974] has
mean ergodic theorems for pseudo resolvents. There is also a local version:
Proposition 1.16. If (λνλ) is a pseudo resolvent in a Banach space X, sup||Af^|| < oo,

84 Mean ergodic theory
and (λη) a sequence with Яп-»оо, then the existence of w-limAn VXrx = у implies
xeX0:= {z: \imx^auXVxz exists}.
Proof, λ νλλη νληχ = λλη(λ — Яп)-1 (VXn — νλ)χ and the boundedness condition shows that
this converges to λη VXnx for λ -> oo. Hence λη VXnx e X0. As X0 is weakly closed, у е X0.
Now
λνλ{Ι-ληνλη)χ = λνλχ-λλη{λ-ληΥ'{νλη-νλ)χ
= (1 + A„(A - Я^ЯЦ* - ЯЯВ(Я - AJ"1 KAn*
tends to 0 in norm for « -» oo. Hence Я Vx (x — y) = 0 and XVxx = XVky ϊον &\\ XAt follows
that χ eX0. D
This also proves that T0x--^lim^^λνλχ satisfies VxT0x = Vxx, which yields Tq = T0.
Similarly, T0 Vxx = ^x. Frequently X = X0 (e.g., for reflexive X). Then the Hille-Yosida
theorem implies that (Я Vx) on Γ0 £ is the resolvent of a semigroup. Pseudo resolvents are
then compositions of a projection on a subspace and a resolvent acting on the subspace.
Proposition 1.16 is taken from Emilion [1984]. Masani [1976] has such a result for
resolvents.
Necessary and sufficient conditions for the convergence of AQtx for all χ have been
given for general strongly continuous semigroups by Lin, Montgomery and Sine [1977].
Yu [1972] has a mean ergodic theorem for "harmonizable" processes.
Cyclic semigroups {Ти}. If Г is a contraction in a Banach space, the sequence
sn = \\Sn(T)x\\ is subadditive. Hence limsjn exists. Derriennic [1976] has shown
that the limit is sup{|<x, h)\: h = T*h, \\h\\ ^ 1}. He also proved lim||rnx||
OO
= sup{\(x,h):he f] T*nS} with S = {he X*: \\h\\ й 1}·
n = 0
A sequence Fn in a Banach lattice is called subadditive for a positive contraction rif Fn+k
•^ Fn + TnFk holds for all n, k ^ 1. The convergence of и-1 Fn has been studied in the paper
of Derriennic and Krengel [1981]. E.g. it is shown that the existence of a weak cluster
point/= 7yof(«"17;)implies||(«"17;-/)-f|| -»O.InLp,(l ^p>< oo), the existence of a
weak cluster point of (n _1 Fn) for non negative (Fn) is equivalent with norm convergence. If
Tis an isometry in Lp, (1 <ρ < oo), and sup {n'1 \\Fn\\p} < oo, then n'1 Fnconverges
weakly, but it need not converge strongly.
If Γ is a contraction and An(T2)x converges for all x, then (almost obviously) An(T)x
converges for all x, but the converse is not true. For an example let X be the Banach space
of sequences χ = (xf: / e Z) of real numbers with ||χ|| = ΣΙ*»Ι<°° and Σ*ί = 0> and ta^e
(Tx)i = xi+1. However, if Γ is positive and X a Banach lattice with order continous norm,
then the converse above does hold; see Derriennic-Krengel [1981]. Sine [1976] has
constructed a uniquely ergodic transformation τ in a compact metric space К (for which
Τ: χ -» χ ο τ is mean ergodic in X = C(K)), such that T2 is not mean ergodic.
Butzer and Westphal [1971] have proved the following (essentially negative) result on
the speed of convergence in the mean ergodic theorem for power bounded linear operators
Γ in a reflexive Banach space X:
\\Anx- Px\\ = o(«_1) implies xe F, and \\Anx- Px\\ = 0(η~χ) implies χ e F0 N. (In
both cases the converse is obvious). For a simplified argument, see Lin-Sine [1983]. For a
strongly continuous bounded semigroup {Tt, t ^ 0} with generator A, Butzer and
Westphal show that \\XRxx — Px\\ = ο(λ), (λ -► 0 + 0), implies χ e ker(,4) = F, and
||ЯЯя.х — Px\\ = 0(λ) implies χ e F® A dom (,4). For related results, see Butzer and
Westphal [1972], Leviatan and Westphal [1973], Leviatan [1974], Goldstein, Radin and
Showalter [1978], and Butzer [1980].

The mean ergodic theorem 85
The papers of Sato [1979], [1979a], Shreider [1967], Rubinov [1977], Lloyd [1976],
and Anzai [1977] are related to Sine's criterion for mean ergodicity of T.
Sato [1975] noticed that Cesaro boundedness can be replaced by Cesaro boundedness
along the subsequence nk with w-lim АПкх = у in theorem 1.1. Emilion [1984] considered
weakened boundedness conditions. E.g., he showed: A positive Abel bounded Г in a
reflexive Banach lattice is mean ergodic. By Heinich [1983] the reflexivity is necessary
here. Emilion's paper also contains an example of Assani of a non positive Cesaro
bounded Г which does not satisfy Tn/n -» 0. For further related work, see Assani [1984a].
π
Ha'inis [1977] remarked that in a Banach algebra η'1 Σyk -> у implies (e + y1/n) (e
ι
+ Уг1п) · · · (e + Уп1п) ~* exp (>>). This yields "multiplicative" ergodic theorems. Sarym-
sakov [1964], [1966] has mean ergodic theorems for vector spaces over a semifield. Atalla
[1976] and Shaw [1983] derived a criterion for mean ergodicity in a Grothendieck space.
General semigroups. Dunford [1939a], and Day [1942] proved mean ergodic theorems
for d-parameter and Abelian semigroups. Koliha [1973], and Nagel [1973] have a
splitting theorem in Banach spaces. We largely followed Sato [1978] in theorem 1.9.
С Ionescu Tulcea [1980] has a generalization of Eberlein's theorem in uniform spaces.
The uniform convexity argument originated with Wiener [1939] and was used by
Garrett BirkhorT[1939] for a single Tbefore L. Alaoglu and G. BirkhorT[1940] obtained their
theorem for general semigroups. Jacobs [1954] proved the existence of a unique fixed
point in co^x for bounded groups Sf in Hubert space and in Lp, (1 <p < oo).
Now consider the space Lx of a finite measure space (Ω, s/, μ). If &" is a semigroup of
measure preserving transformations in Ω and Sf the semigroup of operators Tf=f° τ,
(τ g ^~), then the operators in со Sf are also contractions in L2 с Lu and L2 is dense in Lv
Although theorem 1.10 is not directly applicable to Lu it is not hard to see that Ρ may be
extended from L2 to a projection in Lu also denoted by P, such that Pf belongs, for all
fe Lu ioco^f. (Use the positivity of Γ and the property Л <Ξ D). Ρ again satisfies TP
= PT= Ρ = P2 for all T.
This was proved (for groups) by Aribaud [1970], and extended to more general
operators in Banach lattices by Nagel [1973]. Theorem 1.11 of Nagel [1973] includes
complements due to Kummerer-Nagel [1979].
Affine operators and the equation (/ — 7) дс = у. Iterative methods for the approximate
solution of this functional equation, investigated by Browder, Petryshyn, Koliha, and
others, have motivated the following generalization of Eberlein's theorem by Dotson
[1971]; (having a similar proof):
Theorem 1.17. Let £f — {7^} be any collection of affine operators in a l.c.t.v.s. X, and
{ϋλ, Я g A} a net of affine operators in X which are equicontinuous. Assume ϋλζ eco 6^ ζ for
all λ, ζ. For fixed χ assume
lim (Ux Tux -Uxx) = lim (Tu ϋλχ - ϋλχ) = 0
χ χ
for all Tue £f. If у is a weak cluster point of { ϋλχ} then у = Tuy for all Tu, у е со Sfx and
lim Uxx = y.
Dotson, Groetsch [1975/76], Sato [1979b], and others have given applications.
Lin and Sine [1983] showed for Cesaro bounded mean ergodic Г with so-lim Tn/n = 0
η fc-1
that (i) у e (I— T)X, (ii)xn = η'1 Σ Σ Tjy has a weakly convergent subsequence, and
fc=lj=0

86 Mean ergodic theory
(iii) limx„ = χ with у = (I— T)x, are equivalent. The paper settles the problem of
existence of χ also for dual operators and for many Markov operators.
Krengel and Lin [1984] study the analogous problems for semigroups {Tt, t ^ 0}, i.e.,
the problem when у belongs to the range of the generator A. The necessary condition
t
sup || J Tsyds || < oo is also sufficient for contractions in Lx and for dual semigroups. For
о
α β
mean ergodic semigroups the existence of lima 1J [J Τ5γά8~]άβ is necessary and suffi-
o о
cient. Davies [1982] studies conditions for the range of A to be dense.
Fixed points. There is now an extensive literature on fixed points, see e.g. Istratescu
[1981], Takahashi [1980]. We mention the following fixed point theorem of Ryll-
Nardzweski [1967]:
Theorem 1.18. Let & be a semigroup ofqffine continuous transformations in a l.c.t.v.s. X.
Assume that £f is distal, i.e., for any ιφ^,Ο does not belong to cl {Sx — Sy: S e Sf}. If К is
a weakly compact convex subset ofX with SfKa K, then К contains a fixed point for У.
We refer to Glasner [1976], and Asplund-Namioka [1967] for proofs; see also Day
[1973], Petersen [1983].
Brunei [1973] and Falkowitz [1973] arrived independently at lemma 1.14. We followed
Nagel-Palm-Derndinger [1984]. The lemma also follows easily from the following lemma
of Foguel and Weiss [1973]:
Lemma 1.19. IfPi9 P2 are commuting elements of a Banach algebra with \\ Pl \\ = \\ P2 \\ = 1
andQ = аРг + (1 - α)Ρ2, (0 < α < 1), then \\Qn(P1 - P2)\\ й Kol(\ - а)п~1/2, whereKisa
constant.
§ 2.2 Uniform convergence
1. The uniform ergodic theorem. Let Sn, S, T,... be continuous linear operators
in a Banach space X. The sequence (Sn) is said to converge uniformly (or in the
uniform operator topology) to S, if ||iSn — S|| tends to 0. Г is called uniformly
ergodic if An = An(T) converges uniformly.
This is much more than mean ergodicity. E.g. if X = Lp, and Tis the composition
with a measure preserving transformation, then Τ is uniformly ergodic iff Τ is
periodic with bounded period. Yet, some interesting uniformly ergodic operators
do arise in probability theory and functional analysis. We begin with those results
which do not require spectral theory.
By (1.1) any uniformly ergodic Γ must satisfy
(2.1) ||«_1Γη|| ^0.
As any uniformly ergodic Tis mean ergodic, theorem 1.3 implies that for
uniformly ergodic Τ the space X is a direct sum.F®cl7V with F=F(T)

Uniform convergence 87
= {xeX: Tx = x} and N = (/— T)X. The following theorem, due to Lin
[1974], shows even X = F® N:
Theorem 2.1. If Τ satisfies (2.1), then Τ is uniformly ergodic iff N is closed.
Proof Let Γ be uniformly ergodic. Υ = cl N is invariant under T, and the
restriction S of Γ to Υ satisfies M„(S)|| -* 0. For any η with \\An(S) \\ < l,I-An(S) is
invertible. The invertibility of /— S therefore follows from the identity
(/- S) ("—^ /+ — S + ... + - Sn~2) = I-An{S).
\ η η η J
Hence Υ = (/- 5) У = (/- Γ) Υ a (I- T)X = TV, and Г = TV.
Conversely, if TV equals Y, the open mapping theorem asserts that (/ — T) [/is
open in У for any open U a X. Hence there exists а К > 0 such that, for any у е Y,
there is a ze £ with (7— T)z = >> and ||z|| <; Jf||^||. (Otherwise there is a
sequence % converging to 0 and disjoint to (/— T) {z: \\z\\ < 1}.) From
we see that the restriction S of Τ to Υ is uniformly ergodic. It follows as above
that I-Sis invertible in Y, and (/- T)X = Υ = (/- S) Υ = (/- Τ) Г.
Therefore there exists, for any χ e X, an element у в У with (/— T)x = (I — Г)>>, and
by the invertibility of (/— S) we may assume ||>>|| ^ K'\\{I— T)x\\ with some
К < oo independent of x. We now write χ = {χ — у) + у. As (x — >>) is fixed under
Τ we find
\\An(T)x- (x-y)\\ = ||Λ(Ό>ΊΙ й n-'KK'\\I- Tn\\ ||/- ГЦ ||х||,
i.e., the convergence of An{T) to the projection Ρ with Px = x — у is
uniform, α
It is an exercise to deduce a multiparameter extension: If Tl9 T2,..., Td are
commuting bounded linear operators in X with ||л-1 7JB|| -► 0 and we define Αλ for
λ = (λΐ9..., λά) e Λ = Nd by Лл = ΛΑι(7\)... Αλά{Τά\ then ЛА(к) converges
uniformly for all increasing sequences λ (к) в Л iff (/ — 7J) X is closed for ι = 1,..., d.
(For "only if take kt(k) = к and ЯДА:) = 1, (j Φ /))·
We treat the next criterion right away for a general bounded semigroup 5^ of
linear operators in X. A family {Αλ: λ e Λ} is called a uniformly 5^-ergodic net, if
Л is a directed set, and the Ax9s are linear operators belonging to the norm closure
|| · H-co Sf of со У, such that
(2.2) Χχπνλ\\ΑλΤ-Αλ\\ = § and limj ΤΑλ - Αλ\\ = 0
holds for all ГеУ.

88 Mean ergodic theory
Theorem 2.2. Let {Αλ: λ e A) be a uniformly if-ergodic net for a bounded
semigroup if. Then limA Αλ exists in the uniform operator topology iff there exists a
bounded linear operator Ρ in \\ · \\-mif with TP = PT = Ρ for all Те if.
Proof. If (Αλ) converges uniformly to some P, then obviously Ρ has the desired
properties. Conversely, assume that Ρ exists.
Let AT be a bound for the norms of the operators in if and hence for those in
|| · 11-co 5^. For any ε > 0 we may find a convex combination Ta of finitely many
operators Tte if with || Ta-P\\< s. Clearly TP = Ρ holds even for all Tin || · ||-
co if, and in particular for Τ = Αλ. For large enough λ we have || Αλ Τ{ — Αλ || < ε
and therefore \\ΑλΤα — Αλ\\ <ε. As ε > 0 was arbitrary the estimate \\Αλ — P\\
= \\Αλ-ΑλΡ\\^\\Αλ-ΑλΤΛ\ + \\ΑλΤα-ΑλΡ\\^ε + Κ\\Τα-Ρ\\^(Κ+1)8
completes the proof. D
Remarks. 1) The theorem also follows from theorem 1.9 because if acts on the
Banach space £ = if (X) by left multiplication. 2) The theorem implies that the
uniform convergence of (Αλ) for some net implies the uniform convergence for
any other net. E.g. if Γ is power bounded, the uniform convergence of An(T) for
00
η -+ oo is equivalent to the uniform convergence of (Я — 1) Σ λ~(τη+1)Τηι for
λ il; see also Lin [1974]. m = 0
2. Quasi-compact operators. Kryloff and Bogoliouboff [1937], [1937a]
introduced an important class of operators, for which uniform convergence can be
obtained:
Definition 2.3. Τ is called (weakly) compact if the image under Τ of the unit
sphere of X is conditionally (weakly) compact. Τ is called (weakly) quasi-
compact if there exists an integer m and a (weakly) compact operator Q with
\\Tm-Q\\<l.
We shall use the fact that linear combinations and uniform limits of (weakly)
compact operators are (weakly) compact, and that the product of a (weakly)
compact operator and a bounded linear operator is (weakly) compact. (See e.g.
Dunford-Schwartz [1958]).
Lemma 2.4. Τ is (weakly) quasi-compact iff there exists a sequence Qn of
(weakly) compact operators with \\Tn — Qn\\ -> 0.
Proof. For (weakly) quasi-compact Τ put A = Tm — Q. For any integer η ^ 0
find integers r, s with η = ms + r and 0 rg r < m, and put Qn = Tn — TrAs.
Then we have Qn = Tr[Tms- zf] = Tr[Tms - (Tm - Q)*]. The only term
in the expansion of (Tm — Qf which does not contain Q as a factor is Tms. There-

Uniform convergence 89
fore, Qn is (weakly) compact. \\Δ \\ < 1 implies || Tn - Qn\\ = || TrAs\\
й IIΔ ||s · sup {|| Γ'ΙΙ: 0 й г < т} -* 0. α
The lemma implies that powers of (weakly) quasi-compact operators are
(weakly) quasi-compact. Weak quasi-compactness can be used to prove a mean ergodic
theorem:
Theorem 2.5. Let {Αλ: λ e Λ} be an if-ergodic net for a semigroup if a <£ (X). If
|| · ||-coif contains a weakly quasi-compact operator T, then Px-=\\mxAxx
exists for all xe X, and Ρ is weakly compact. If Τ is even quasi-compact, then Ρ is
compact.
00
Proof. Again put Δ = Tm - Q. It is clear from \\Δ \\ < 1 that {I- Δ)~ι = Σ Ak
exists, and it is simple to verify the identity k = 0
(2.3) Ax = (/- AY'QAX + (Ι- ΔΓΗΑχ - TmAx).
Fix any x. By the definition of an 5^-ergodic net \\ναλ{Αλ — SA^x = 0 holds
for all Seif, and hence for all 5e||· ||-соУ\ Powers of elements of zoif
belong to со if. Therefore Те || · ||-co if implies Tm e || · ||-co if, and we obtain
limA {Αλ — ТтАл)х = 0. By the weak compactness of (/ — Δ)~Χ Q and the boun-
dedness of {Αλχ: Xe Λ) there exists a weak cluster point of (/— Δ)~χQAxx. It
then follows from (2.3) that Αχχ has a weak cluster point. By theorem 1.5 this is
enough to prove convergence.
As the net is 5^-ergodic SP — Ρ holds for all S e if. If Taz is a convex
combination of elements ^ ζ with Ste if, we therefore have TaPz = Pz. As ^z belongs to
со if ζ for all z, we obtain ΑλΡ = P, and a similar argument shows Tm ΑλΡ = P.
Now use Ρ = ΑλΡ and (2.3) to prove Ρ = (I — Δ)'1 QAXP. The (weak)
compactness of Q yields the (weak) compactness of P. α
If if is an arbitrary bounded semigroup such that || · ||-co if contains a quasi-
compact T, an application of theorem 2.5 to An(T) shows that F(T) and hence
also the fixed space of if is finite dimensional: We have F(T) = {x: Px = x}
= i>£and {xeF(T):\\x\\ S 1} = P{x: II x II й 1} is compact.
Theorem 2.6. Let {Αλ: λ e A) be a uniformly if-ergodic net for a semigroup if. If
|| · ||-co5^ contains a quasi-compact T, then \\Αλ — P\\ tends to 0.
Proof. With the notation used above Q is now compact. Using Tm = Δ + Q and
00
{Ι— Δ)~ι = Σ Δ* it is simple to verify
о
(2.4) Αλ = AkQ(I- Δ)'' + {Ak- AxTm){I- Δ)''.
As the net is uniformly 5^-ergodic || Ax — AXS\\ -*■ 0 holds for all Se Sf and

90 Mean ergodic theory
hence also for S = Tme || · ||-co5^. It is therefore sufficient to show that
Αλζ){1— Δ)~ι converges uniformly, or that Αλ converges uniformly on the
compact set C:= cl (£>(/- Ay1 {x: \\x\\ < 1}). Let К be a bound for all \\Αλ\\ and for
11Ρ11. For any ε > 0 there exist finitely many cl9 ...,cke С such that any ze С has
distance rg ε/Κ from some c(z) e {c1?..., ck}.
As Н^дХ — Px\\ -* 0 holds for each χ by theorem 2.5, we have
\\Axci — Pci\\ <s for i = !,..., к for large enough A. Then \\Αλζ — Pz\\
й \\Axc(z)\\ + \\Axc(z) - Pc(z)\\ + \\Pc{z) - Pz\\ < 3ε holds for all ze C. α
Theorem 2.5 and 2.6 are Eberlein's [1949] generalizations of results of Yosida-
Kakutani [1941], who treated the case Αλ = Αη{Τ). Yosida and Kakutani
proved even a structure theorem for power bounded quasi-compact Τ (theorem
2.8 below). Its proof depends on some spectral theory.
We refer to Dunford-Schwartz [1958] for an introduction to spectral theory, but
we summarize the main definitions here for the convenience of the reader:
Let ГЬе a bounded linear operator in a complex Banach space X. The resolvent
set q(T) is the set of complex numbers λ for which the resolvent R{X, T) —
(λΐ— Τ)~ι exists as a bounded linear operator with domain X. The function
λ -* i?(A, T) defined on q{T) is holomorphic. (The theory of vector valued
holomorphic functions is analogous to that of complex valued holomorphic
functions). The spectrum σ{Τ) -.= C\q(T) is a non empty compact subset of C.
r (T) := sup {| λ |: λ e σ(Τ)} is called the spectral radius of T. It may be computed
from r{T) = lim||rw||1/n. For |Я| > r(T) we have i?(A, T) = Σ Τη/λη+1.
и = 0
λ0 is called a pole of order и, {η ^ 1), if i?(A, T) has a Laurent expansion
Д(Я,г)= Σ Β,(λ-λ0)\ (ΰ_„ + 0)
к=-п
near λ0. B_x is the residue of i?(A, Г) at A0.
We can now continue with our study of uniform ergodicity:
Theorem 2.7. A power bounded linear operator Τ in a Banach space X is uniformly
ergodic iff either 1 belongs to the resolvent set ρ (Τ) or 1 is a pole of first order of the
resolvent /?(Я, Т).
Proof If Τ is uniformly ergodic N = (I— T)X is closed and X = F® N by
theorem 2.1. If S is the restriction of Γ to TV, and Ρ the projection of X onto i% we
can verify
(2.4) i?(A, Γ) = [(A - Ι)"1 Ρ + i?(A, 5) (/- P)] for 1 Φ Я е ρ(5)
by showing that (λΐ — Τ) [... ] χ = χ holds both for χ e F and for χ e N. We
know from the proof of theorem 2.1 that /— S is invertible. Thus 1 belongs to

Uniform convergence 91
ρ (iS), and R (Λ, S) is holomorphic in a neighborhood of 1. In the case F Φ {0}, the
projection Ρ is non degenerate and 1 is a pole of first order of i?(A, Γ). In the case
F = {0} we have X = TV and Γ = S and 1 belongs to ρ (S) = ρ (Τ).
Conversely, the uniform ergodicity of Τ follows in the case 1βρ(Τ)
because then the invertibility of/— Γ shows that (I — T)X is £, and hence closed.
Finally, if 1 is a pole of first order of i?(A, Γ), we have an expansion i?(A, T)
00
= Σ Βη (λ — 1)" for 1 Φ Я in some neighborhood of 1. Spectral theory asserts
n=-l
that !?_! is a projection onto i7. Let S be the restriction of Г to Υ — clN.
For Я with |Я| > 1 ^ r{T) ^ r(S), i?(A, S) is just the restriction of Д(Л, Г)
00
= Σ TkjXk+1 to У. As jB_x and Г commute, we have B_x Tx = Г5_хх = 5_xx
fc = 0
for all x. Therefore the restriction of 5_х to evanishes and i?(A, S) has the form
00
Σ Д,(Л — 1)" for λ with | Λ | > 1 in a neighborhood of 1. This implies that 1
n = 0
belongs to the resolvent set of S, and Υ = (I — S) Υ Thus Υ is contained in
(I— T)X = N, and TV = Υ is closed. By theorem 2.1 Γ is uniformly ergodic. α
If Γ is power bounded and quasi-compact, then ξ Τ is quasi-compact, and hence
uniformly ergodic for each ξ on the unit circle. It therefore follows from i?(A, T)
= (λΐ- ТУ1 = ξ(ξλΙ-ξΤ)-1 = £Κ(ξλ, ξΤ) that Д( Τ) can have only poles
of first order on the unit circle, and no other singularities. Consequently, there
can be only finitely many poles λΐ9..., Хк with | Д. | = 1. (There may be none). λ{ is
a pole of i?(., T) iff 1 is a pole of i?(., Яг1 Τ). Set ?: = Яг1 Г. The proof of
theorem 2.7 shows that the residue B_x (7J) in the expansion of R(., 7J) in a
neighborhood of 1 is the projection of X onto the fixed space F(TJ), and that A£ is a pole
iff ^(TJ) = {x: Tx = /^х} differs from {0}, i.e., iff A£ is an eigenvalue of Г.
As T{ is quasi-compact, ^4„(7J) converges strongly to a projection Pt with fixed
space F(7J) and with ЗД = i^ = P( = if. F(7J) is finite dimensional by the
remark after Theorem 2.5. As Am(T^ and An(Tj) commute, the projections Pt
commute, too. From P^x = P}Ptx e F(1D η F(Tj) we infer that PtPj = 0 holds
for i Φ j. Set
к
s*=t- Σ hPi.
i = l
Using the relations above we find PtS = ST* = 0 for ι = 1,..., k. We can now
state the uniform ergodic theorem of Yosida-Kakutani:
Theorem 2.8. Let Τ be a power bounded, quasi-compact linear operator in X. Then
each power Tn has a representation
(2.5) Tn= Σ А?Р£ + 5И
i = l

92 Mean ergodic theory
with Pb S as above. There exist constants 0 < ρ < 1, Μ > 0 with \\Sn\\ rg Μρη,
(и = 1,2,...).
Proof. (2.5) follows by induction from the definition of S. As the terms Х}Р{ are
norm bounded, S is power bounded. Since the range F(fy of Pt is finite
dimensional, Pt is compact. Hence S is quasi-compact. If χ is an eigenvector of S with
eigenvalue λ having moduls 1, then λ2 χ = S2x = TSx = TUx follows from 7*5
= 0. Я must then be some λχ and χ e ^(7^) but then Я,χ — Sx = Tx — XtPtx = Ягх
— Я, χ = 0 yields a contradiction. Thus Slias no eigenvalues of modulus 1 and the
entire unit circle belongs to the resolvent set of S. As S is power bounded we have
r (S) rg 1 and therefore r (S) < 1, which is equivalent to the existence of Μ and ρ in
the statement of the theorem, α
N-l
Remarks. As TV"1 Σ Λ" tends to 0 for A£ Φ 1, the uniform convergence of AN (T)
n = 0
follows. (We did not use theorem 2.6). The definition of/· and S above could be
replaced by an application of the spectral projections corresponding to the
spectral sets UJ,..., UJ, σ(Τ)\{λΐ9..., AJ.
If Γ is a linear operator having a representation as in theorem 2.8, Γ clearly is
quasi-compact. Looking at the proof we therefore see that a power bounded Τ is
quasi-compact iff {ze σ(Τ): |z| = 1} containes only poles of i?(A, T) and the
corresponding eigenspaces are finite dimensional.
The study of quasi-compact operators was largely motivated by applications
to Markov operators, which will be treated in the supplement on Harris
processes.
Notes
Lin [1974a], [1977] has also characterized uniform ergodicity for strongly continuous
semigroups {Tt, t ^ 0} of bounded linear operators in X with T0 = / and || Tt/t\\ -» 0,
(/ -► oo). With the notation of the previous section the following conditions are shown to
be equivalent:
(i) There exists a bounded operator Ρ with \\A0tt — P\\ -+ 0, (t -+ oo);
(i) the infinitesimal generator A has closed range;
ι
(iii) S0tl = J Ttdt is uniformly ergodic;
о
(iv) there exists a projection Ρ on {x: Ttx = xVt > 0} with \\XRX - P\\ -> 0 for λ -> 0;
(ν) (λ Rx)n converges uniformly for some (every) λ > 0.
This generalizes results of Hille and Phillips [1957].
Dunford-Schwartz [1958] proved a uniform ergodic theorem for quasi-compact Twith
wo-lim Tn/n = 0.
Brunei and Revuz [1974a] proved that any bounded linear operator is quasi-compact
iff it is the sum of an operator S with r(S) < 1 and an operator Fwith finite dimensional
range.
An interesting class of quasi-compact operators in spaces of Lipschitz functions has

Uniform convergence 93
been introduced by C.T. Ionescu Tulcea and G. Marinescu [1950]. They consider a
Banach space X which is a subset of a Banach space 9). The norm ||| · ||| in X may differ
from the norm || · || in 9), but it is assumed that xn e X, \\\xn\\\ ^ K, \\xn — x\\ -> 0 implies
xeX and 111 χ \ \ | <Ξ K. 11 is proved that any bounded linear operator Τ in X with the
following three properties is power bounded and quasi-compact: (i) sup {\\Tnx\\: ne N, xeX,
\\x\\ ^1}<οο; (ii) there exists R>0 and 0<r<l such that |||7x||| ^r|||x||| + Д]MI
holds for all xeX, (Hi) the image under Г of any ||| · |||-bounded subset of X is
conditionally compact in 9).
This axiomatic formulation is fitted for the following situation: 9) is the space of
continuous functions on a compact metric space (C, ρ) and \\y\\ = sup {|y(c)\: с e C}. X is the
subspace of functions x, satisfying, for some 0<d^l, a Lipschitz-condition L(x)
= sup{\x(t) — x(s)\ g(t,s)~d: t Φ s} <oo, and |||x||| is given by ||x|| + L(x). We assume
that (W, Ψ*) is a measurable space, and P: (c, A) -» P(c, A) is a given map С χ iV -> U,
which is Borel measurable for any fixed A, and a signed measure of total variation ^ 1 for
any fixed c. If τ: С χ W -» С is a map with ρ (τ (с x, w), τ (c2, и>)) й г Q (с х, с2), which is Borel
measurable for any fixed c, we can define Τ by
(Tx) (c) = J x(t(c, w)) P(c, dw).
w
Such operators have originally been introduced for the study of chains with complete
connections, but there are now also other applications; see e.g. Hofbauer-Keller [1982].
Positive operators. We now consider positive (necessarily bounded) linear operators in
the complexification X of a Banach lattice. For example X may be a complex Lp-space.
Karlin [1959] proved that Τ is uniformly ergodic iff r(T) 5Ξ1 holds and 1 is a pole of
R(X, T) of order <Ξ 1, (i.e., 1 e q(T)ov 1 is a pole of order 1). Another equivalent condition
is: r(T) ^ 1 holds and (λ - 1) Д(А, Τ) converges uniformly for λ j 1; (cf. Lotz [1981]).
The spectral theory of positive operators has profound implications for the relation of
uniform ergodicity and quasi-compactness. The key result is a theorem of Niiro-
Sawashima [1966] in the form of Lotz and Schaefer [1968]: If r(T) = 1 is a pole of
R(X, T) and the range of the residue Ρ of R (λ, Τ) at λ = 1 is of finite dimension k, then
{λ e σ(Τ): \λ\ = 1} consists entirely of poles of R(X, T). We refer to Schaefer [1974: 331]
for the proof. Lin [1978] showed for the same class of operators that the range of the
residue in each pole on the unit circle is of dimension <Ξ k. It then follows that for positive
Τ the following conditions are equivalent:
(a) T is uniformly ergodic and F(T) is finite dimensional;
(b) Τ is quasi-compact and {(λ — 1) R (λ, Τ): λ > 1} is uniformly bounded;
(c) Τ is quasi-compact and n~lTn converges to 0 in the weak operator topology.
[The positivity of Τ is crucial: There exists a uniformly ergodic non positive contraction Τ
in (2 w^h dim (F(T)) < oo, which is not quasi-compact: If el9 e2,... is an ON-basis and
λΐ9 λ2,... a sequence on the unit circle converging to Я φ 1, then Τ can be given by Tek
= ккек.~]
Lotz [1981] has explored conditions for uniform ergodicity of positive linear operators
Tin C(X) with 71 = 1 (Markov operators), where Xis a compact Hausdorff space. In the
following theorem he gives sufficient conditions for quasi-compactness which are also
necessary when Τ is irreducible.
Theorem 2.9. A Markov operator Τ on C(X) is quasi-compact if the following conditions
(i)-(iii) are satisfied:
(i) T* is mean ergodic,
(ii) F(T*) has a weak order unit, i.e., there exists α μεΕ(Τ*) with ν <ζ μ for all
ν e F(T*);

94 Mean ergodic theory
(in) every probability μ e F(T*) has поп meager support.
This has several interesting consequences. E.g. using the representation of L^ as a space
C(X) one can deduce that every positive irreducible contraction Τ in L^ is uniformly
ergodic. (ris called irreducible if there exists no closed Γ-invariant ideal. An ideal is a
subspace /with/e /, \g\ <Ξ \f\ => ge I. Ando [1968] had proved that such Гаге mean
ergodic).
Apparently the peripheral spectrum σ(Τ) η {λ e С: | λ\ = r(Τ)} is particularly
important for questions of uniform ergodicity. For this we refer to Schaefer [1974]. Lotz [1968]
proved that positive operators Tin a Banach lattice have cyclic peripheral spectrum if they
satisfy a growth condition, satisfied, e.g., when (λ — r(T)) Κ(λ, Τ) is uniformly bounded
for λ > r(T).
Uniform ergodic theory for positive operators in Banach lattices is also studied in the
thesis of Axmann [1980] and in the book of Nagel-Palm-Derndinger [1984].
Deshpande and Padhye [1979] showed: If {Tt, t ^ 0} is a semigroup of self-adjoint
operators such that 7i is compact and 1 e σ(7ί), then Tt converges in norm for t -» oo.
We also mention that Istratescu [1974], [1977] has considered generalizations of the
class of quasi-compact operators using measures of non-compactness. R. O'Brien [1978]
has an application of quasi-compact operators to control theory.
Deshpande and Padhye [1977] have proved ergodic theorems for quasi-compact
operators in barrelled spaces.
§ 2.3 Weak mixing, continuous spectrum and multiple recurrence
We first prove the spectral mixing theorem for contractions Γ in a complex
Hubert space X: The vectors χ in the orthogonal complement Xfl of the closed
space Xuds spanned by the eigenvectors belonging to eigenvalues λ with modulus
n-l
IA| = 1 are exactly those which satisfy n~l Σ |<7T,'x,y}| = 0 for all yeX. The
i = 0
results for general Τ will be applied to the isometries arising from endomorph-
isms in a probability space. We obtain the principal results about weak mixing.
Finally, we prove Furstenberg's ergodic theorem for generic measures and give a
sample of an application.
The splitting of X into a direct sum of a space Xuds on which Thas "unimodular
discrete spectrum" and a space Xfl of "flight vectors" will be extended to more
general Banach spaces and to more general semigroups in the next section. But
here the tools are simpler. Actually, further obvious simplification is possible for
the case of isometries, which is sufficient for the study of endomorphisms.
A sequence (an)neZ of complex numbers is called поп negative definite if
Σ/ HmziZmai-m^Q holds for every finite sequence {z{. \l\un) of complex
numbers. Here ζ is the complex conjugate of z. Put
Tn = T\ (n ^ 0), and Tn = Γ*""', (n < 0).
Proposition 3.1. For any contraction Tin a Hilbert space X, and for any xeX, the

Weak mixing, continuous spectrum and multiple recurrence 95
sequence (Tnx,x} is поп negative definite. There exists a unique finite measure vx
on R/Z = [0,1 [, the spectral measure of x, with
< Tn x9 x> = J exp (2 π int) vx (dt), (neZ).
Proof. The second assertion follows from the first and a well known theorem of
Herglotz [1911], sometimes called Bochner's theorem. The first assertion is
simple when Τ is an isometry:
η
Σ ZlZm\Ti-mX, X/ = Σ ΣζΐΖτη\Τι + ηΧ> Tm + nX)
1,т= —η Ι m
= \\Σ^,τι+ηχ\\2^ο.
Ι
As the first identity in this calculation may fail for contractions, the proof in the
general case involves more effort. For 0 < r < 1 and / e [0,1 [ put
U(r,t)= Σ rkQxp(2nikt)Tk
fc^O
and
V(r, t) = Σ r|k| exp (2nikt) Tk = -I+ U(r, t) + U(r, if.
fceZ
With у = £/(r, i)x, \\y — x\\ й \\y\\ follows from у — χ = r exp (2nit) Ту, and
this yields
<F(r, t)x, x} = - <x, x> + <>>, x} + <χ,>>>
For complex {z{. \l\<Ln) and 0 < r < 1 we then obtain
Σζ^-^^Τ^χ,χ}
l,m
1
= Σ Σ ZiZmrlkl <Tkx, x} $ exp (2ni(l - m - k)t)dt
l,m к 0
= \ΣζιΖγη exP (2πι (/ — m) 0 < ^(/, 0*> *> ^
= J ΙΣ^ι exP (2nilt)\2 <K(r, f)*, *>Λ ^ 0.
When r tends to 1 the desired inequality follows, α
The upper density D * (M) of a subset Μ с Ν υ {0} is defined by
Z>* (M) = lim sup n~l card (In [1, ή])
and the lower density D+ (M) is the corresponding lim inf. We say that Μ has
density D(M) if D*(M) = ДДМ) = D{M). A sequence (tfw)n^0 is said to
converge in density to a, if there exists a subset McN with D(M) = 0 and

96 Mean ergodic theory
lim^^,Пфмап — α· Equivalently, for any ε>0 the set of integers η with
\an — a\ > ε has density 0, We then write D-lim an — a. It is an exercise to prove
that for bounded sequences (an) one has
n-l
(3.1) D-lim an = а о \imn~1 Σ \an — a\ = 0.
k = 0
Lemma 3.2 (Wiener). Let ν be a finite measure on [0,1 [ and
v(n) = Jexp (Inint) v(di) its Fourier transform. Then ν (ή) converges in density to
Oiffv has no atoms, i.e., v({/}) = 0 for all t.
Proof ν (ή) converges in density to 0 iff ν (ή)2 converges in density to 0. We have
n-l n-l
n~x Σ \v(k)\2 = n~1 Σ iexp(2nikt)v(dt)$exp(-2niks)v(ds)
fc = 0 k = 0
n-l
= ]Ί>_1 Σ exp(2nik(t-s)y]v(dt)v(ds).
k = 0
The last term tends to
j ν(Λ)ν№)= Σ Hit})2
{t = s} 0^f<l
as η -* oo because the integrands [... ] are everywhere bounded by 1 and
converge to l{i=s}. α
Lemma 3.3 (Akcoglu-Sucheston). IfS is a contraction in a Hilbert space X, then
\\x\\2 — \\Sx\\2 < β2 implies that for each yeX
(3.2) \<x,y}-(Sx,Sy}\ue\\y\\.
Proof We have
\<x,y> - <Sx, Sy}\ = |<*,j> - <S*Sx,y}\ й \\х- S*Sx\\ \\y\\.
Therefore (3.2) follows from ||jc — S*Sx\\2 = \\x\\2 - 2\\Sx\\2 + ||S*Sx||2
^||χ||2-||5χ||2^ε2. D
Theorem 3.4 (Spectral mixing theorem of Koopman-von Neumann [1932]). Let
Τ be a contraction in a Hilbert space X. Then the following statements about an
element xeX are equivalent:
(a) x e Xfl;
(b) vx has no atoms;
(c) D-lim <Γηχ,χ> = 0;
(d) D-lim (Tnx, y} = 0 for all yeX.
Proof (a) => (b): Let xe Xfl and assume vx({t}) > 0 for some /.

Weak mixing, continuous spectrum and multiple recurrence 97
Applying von Neumann's mean ergodic theorem to the contraction
exp ( — 2nit) Γ we find that the averages An(exp( — 2nii) T)x converge to an
element у e X with exp ( — 2πίί) Ту = у. We have
{у, χ} = lim η'1 Σ exp( — 2nitk) {Tkx,x}
и->оо fc = 0
= lim ]Ή-1 Σ exp(2ni(s-t)k)vx(ds) = vx({t})>0.
и->оо fc = 0
Hence >> φ Ο, and у is an eigenvector with eigenvalue exp (2nii) which is not
orthogonal to x. This contradicts our assumption.
(b) => (c) follows from Lemma 3.2.
(c) => (d) By (3.1) the set of elements yeX with D-lim <Γηχ,>>> = 0 is a
closed linear subspace of X. For every у which is orthogonal to all Tkx, (к ^ 0),
we even have <Гих, у} = 0. It therefore remains to show D-lim (Tnx, Tkx) = 0.
As || Tnx\\ decreases we have || Tnx\\ - \\ Tn+kx\\ < ε2 for all large n. Lemma 3.3
then implies \(Tnx, x} - (Tn+kx, Tkx}\ < ε.
As ε > 0 was arbitrary, (c) yields D-lim„ (Tn+kx,Tkx} = 0, which is
equivalent to D-limn<rnx, Tkx} = 0.
(d) => (a) If w is an eigenvector of Τ having an eigenvalue λ with | λ | = 1, then
n-l
S = λ -1 74s a contraction and wis fixed under S.(d) saysthatH-1 Σ К^%.У>1
i = 0
tends to 0 for all y. Because of | (S*lx, y} \ = |< Tl x, у} | this implies the weak and
therefore also the strong convergence of An(S)x to 0. By theorem 1.3 and lemma
1.1.2, χ is orthogonal to w = Sw. As w was arbitrary we obtain
The terminology "spectral mixing theorem" is motivated by the application to
endomorphisms τ of a probability space (Ω, sf, μ), τ is said to be weakly mixing iff
(3.3) D-lim μ(Αητ~ηΒ) = μ{Α)μ{Β)
л->оо
holds for all ,4, Be si.
Comparing (3.3) with proposition 1.1.8 we see that weak mixing implies
ergodicity. The isometry/-» Tf = for inX = L2 always has the constant
function 1 as an eigenvector with eigenvalue 1.
Theorem 3.5. For an endomorphism τ of a probability space (Ω, s/, μ) the
following conditions are equivalent:
(α) τ is weakly mixing;
(b) O-limn(Tnfg} = </, 1> · <bg>for allfgeL2;
(c) D-limn<rnA,A> = 0>ra//Ae{1}1;
(d) Τ has continuous spectrum (i.e., no eigenvectors) in {H}1.
Proof. (3.3) is equivalent to the validity of (b) for/= 1A and g = \B. Thus (b)

98 Mean ergodic theory
implies (a). Approximating general/, g e L2 by linear combinations of indicator
functions we also obtain (a) => (b). (b) => (c) is trivial. As no eigenvalues of
modulus less than 1 exist, the spectral mixing theorem shows that (c) implies (d),
and that (d) yields D-lim <7T"/', g'} = 0 for all/, g' e {1}1. To prove that this
implies (b), put/ =/- </, 11 >, g' = g - <g, 1 > and check
<7in/^) = <r/4</J)l^ + <gJ>1> = <r/^/) + </,1)<ig). □
Another useful characterization of weak mixing can be given via direct products
of endomorphisms: If τί? (ι = 1, 2) are endomorphisms of probability spaces
(Ωί? j2/f5 /if) we define the direct product τχ χ τ2 in the product
(Ωχ χ Ω2, s/t ® «я/2> A*i x Hi) by
τι χτ2(ω1?ω2) = (τ1ω1,τ2ω2).
It is easy to check that τχ χ τ2 is an endomorphism. The definition of
τχ χ τ2 χ ... χ τη is analogous.
Theorem 3.6. fa,) Ifxx is weakly mixing and τ2 ergodic, then τχ χ τ2 £y ergodic,
(b) Ι/τχ and τ2 are weakly mixing, then τχχτ2 is weakly mixing;
(c) If τ χ τ is ergodic, then τ is weakly mixing.
Proof, (a) Put ΐ = τ1χτ2,/ϊ = μ1χ μ2. For any Al9 Bx e stfx and A2, B2 e stf2
we have
μ((Αχ χ ^ητ"^^ x B2)) = μ1(Α1ητϊίΒ1)μ2(Α2ητ2ίΒ2).
The first factor on the righthand side tends to μ1(Αί)μ1(Β1) except along a
sequence of density 0. Taking Cesaro averages, and using
M^OM^i) Иг<Аг> ^№) = μ(^ι x Л2) Д(51 x #2)
and „_!
л * Σ μ2(Α2ητ2ίΒ2) -* μ2(Α2)μ2(Β2)
i = 0
we find that
(3.4) /Γ1^ μ(Λητ-*β) -* μ(Α)μ(Β)
i = 0
holds for sets of the type ^ = ^ χ ^42, 5 = jBx χ 2?2. But then (3.4) remains true
for finite disjoint unions A, B of such "rectangles" and, by approximation, for all
Я, Be si\® stf2. Thus, f is ergodic.
(b) Apply the same argument to the convergence in density of А(Апт~1В)
and observe that finite unions of sequences of density 0 have density 0.
(c) τ must be ergodic. Assume/ Φ 0 is an eigenvector of Γ, then the eigenvalue
has modulus 1 because Γ is an isometry, and thus |/| is τ-invariant and hence

Weak mixing, continuous spectrum and multiple recurrence 99
constant. Now #(ω1? ω2) =/(ω1)//(ω2) satisfies #(τω1? τω2) = λ/(ωί)/λ/(ω2)
— g (ω1? ω2). As τ χ τ is ergodic g is constant on Ω χ Ω and hence/is constant on
Ω. As the only eigenvectors of Τ are constants τ is weakly mixing by theorem
3.5. α
A glimpse into Furstenberg's multiple recurrence theory. The defining properties
of mixing and weak mixing involve only pairs of sets. Similarly, ergodicity is
characterized by the convergence of the Cesaro averages of μ {Α η τ~'Β) to
μ(Α)μ(Β) for all pairs A, B. It is not clear how this could be used for a proof of
limiting results for expressions involving three or more sets. While the question
whether mixing implies μ (Α η τ "'Β η τ " 2i С) -* μ (Α) μ (Β) μ (С) remains
unsolved, the multiple recurrence theory yields
(3.5) D-lim μ(Α0 η τ^ηΑ1 ... ηтк~пАк) = μ(Α0)... μ(Αϊί)
η
for commuting weakly mixing τ1?..., тк. The Furstenberg-Katznelson theorem
asserts
n-l
(3.6) liminffl"1 Σ μ(Α0ητϊίΑ1η ...пт^1Ак) > О
η i = 0
for arbitrary commuting automorphisms τΐ9..., тк of a probability space and
arbitrary yl0?..., Ak of positive measure. Furstenberg has shown that results like
(3.6) are equivalent to deep theorems in combinatorial number theory. E.g. the
case τ£ = τ1 yields a theorem of Szemeredi, saying that any subset Af с 1Ч1 with
positive upper density contains arbitrarily long arithmetic progressions.
As the recent monograph of Furstenberg [1981] gives a beautiful systematic
account, a brief introduction shall suffice here. We begin with Furstenberg's
ergodic theorem for generic measures.
Definition 3.7. Let τ be an endomorphism of a probability space (Ω, я/, μ). If if
is an algebra of bounded, complex valued, ^-measurable functions which is
closed with respect to complex conjugation and satisfies g ° τ e У? for all g e У?,
we call a probability measure ν on (Ω, jtf) generic with respect to if in (Ω, «я/, μ, τ)
when
(3.7) iAn(T)fdv-+ifdl*
holds for all fe <£.
The measure ν need not be invariant! The terminology is motivated by an
example in topological dynamics: If τ is a continuous transformation of a
compact metric space Ω, a point ω is called generic for μ if (Anf) (ω) tends to \/άμ for
all/in the algebra if = C(Q). Thus ω is generic iff the point mass in ω is generic.

100 Mean ergodic theory
Forfe У? also |/|2 belongs to !£ because if is a conjugation closed algebra. If/
is τ-invariant, (3.7) shows
n-l
Σ
i = 0
f |/|2rf|i = limn-1 Σ f l/l2oTlrfv = f |/|2rfv.
The L2(//)-norm ||/||2>μ then agrees with ||/||2tV. If & contains a set of
τ-invariant elements of L2 (μ) which is dense in the space L2 (μ,./,) of τ-invariant
elements of L2 (μ), then this subset and therefore all of L2 (μ, Уt) is embedded
isometrically in L2(v). Let Ρτ be the projection in L2(ju) onto ^г(Л «^t)·
Theorem 3.8. Let ν 6e generic with respect to if ш (Ω, si, μ, τ). Assume that if
contains a dense subset ο/Σ,2(μ, </τ). Тйел Anf tends to PJin the norm || · ||2,v of
L2(v)forallfe&.
Proof. Let ε > 0 be given. Let g be an element of !£ nL2 (μ, </τ) with
\\Ρτ/— ε\\ι,μ <ε· As the inequality is strict, von Neumann's mean ergodic
theorem and the τ-invariance of g imply the existence of a AT with
ί IAK(/— g) |2 φ < ε2. Put A = /— g. By the properties of !£ the function
2
\AKh\2 again belongs to if. For all large и, (3.7) therefore yields
$AH\AKh\2dv<e2.
n-l n-l
Applying the Cauchy-Schwarz inequality (л-1 Σ α£)2 ^ л"1 Σ α? to af
i = 0 i = 0
= | AKh Ι (τ*ω) we find | AnAKh \2 rg An \ AKh |2, and hence J| AnAKh \2dv < ε2. For
large η the function AnAKh is close to Anh\ Indeed, when о 0 is a bound for A,
we have
\п-1к-1^ Σ Α°τί+;-72-ιηΣ ЛотЧ^гл^с.
i = 0 j = 0 1 = 0
Consequently, \\Anh\2dv < ε2 holds for large enough n. Now
Mn/-^/H2,v<2e follows from \\Anh\\2fV = \\Anf-g\\2fV < ε and
The example of chief interest is the diagonal measure in a product space. Let τ be
an endomorphism in a probability space (Ω, .я/, μ). Let σ be the product τ χ τ2
in (Ω2, ja/2, μ2) = (Ω χ Ω, si ® л/, μ χ μ). The diagonal mesure μ^ on si2 is the
unique measure with
№>ι, ω2)άμΔ{ωΐ9 ω2) = ]Υ(ω1? ω1)άμ(ω1)
for all bounded ja/2-measurable/: Ω2 -* R. For/, g e Loo (μ) we denote by/® g
the function on Ω2 with {f®g) (ω1? ω2) = /(ω1)#(ω2). Let
if = L^(μ) ® L^(μ) be the algebra of finite sums of such functions/® g.
Proposition 3.9. If τ is ergodic, then μΔ is generic for !£ in (Ω2, si2, μ2, σ).

Weak mixing, continuous spectrum and multiple recurrence 101
Proof. J5? has the properties required in definition 3.7, and
n-l n-l
\n~l Σ (f®g)0<ridμA = $n-1 Σ/(τίω1)^(τ2ίω2)^(ω1,ω2)
i = 0 i = О
= "_1 Σ* |/(τ'ω)£(τ2'ω)</μ(ω)
i = 0
n-l
= и-1 Σ \f{ω)g{τiω)dμ{ω)
i = 0
^(\fdμ){\gdμ) = \(f®g)dμ2. Π
A sample application shall illustrate the spirit of the method: If τ is weakly
mixing, σ is ergodic, and L2 (μ2, У σ) consists only of the constants. In particular,
For/, ge Loo (μ), Theorem 3.8 implies the strong convergence in L2(/0 °f
Αη(σ) {f® g) to Pa{f® g). As strong convergence implies weak convergence,
this yields
<Α®1,^(σ)(/'®ί)>-><Α®1Ι,Ρσ(/'®ί)>
for h e Loo (μ), or more explicitly
(3.8) η^^Σ ЬШЬ^(?2*<»№ - (fArf|i)(f/rf|i)(fgrf|i).
We apply this result to τ and h = 1 Α,/= 1B? g = lc? and also to the weakly mixing
transformation τ χ τ with h = 1A ® lA,/= 1B ® 1B? g = lc ® lc> and obtain
n-l
и"1 Σ μ(Αητ-ίΒητ~2ί€) -* μ{Α)μ{Β)μ{0)
and ί=0
n-l
и"1 Σ μ2 О* х^пГ!Вхт-^т-21*Схт-2Т) -* (μ (Λ) μ (5) μ (С))2.
ί = 0
If we put at = μ(Αητ~ίΒητ~2ί€) and α = μ(Α)μ(Β)μ(€), we therefore
n-l n-l n-l n-l
have n~l Σ α? -* α2 and и-1 Σ α,· -* α. Then η~χ Σ (^ - α)2 = η'1 Σ α?
ί = 0 ί = 0 ί = 0 ί = 0
η-1
— 2 Σ α^α + α2 tends to 0. By (3.1) this means D-lim (af — a)2 = 0 and hence D-
i = 0
lim(af — a) = 0.
We have proved
Theorem 3.10. If τ is a weakly mixing endomorphism we have
lim^-1^ \μ(Αητ-ίΒητ-2ί€)-μ(Α)μ(Β)μ(€)\ = 0
i = 0
for all A, B,C est.

102 Mean ergodic theory
(As even (3.5) can be handled without the generic ergodic theorem, its main use is
the deeper ergodic case).
Notes
If Tis a contraction in a Hubert space X, then 3E also decomposes into the direct sum of Xd,
the closure of the space spanned by all eigenvectors, and its orthogonal complement Xc. Τ
is said to have discrete spectrum if X = Xd and continuous spectrum if £ = 3£c. For isome-
tries we clearly have Xd = Xuds and Xc = Xfl. These subspaces have also been characterized
geometrically: Call 0ф/е£ a weakly wandering vector, if there exists a sequence
k1 < k2 < ... of integers such that the vectors Tkif are orthogonal to each other. It is
simple to see that for any isometry Τ the weakly wandering vectors are orthogonal to Xd.
Krengel [1972] has proved that they are dense in Xc. Xc therefore is the closed subspace
spanned by the weakly wandering vectors. An alternative (unpublished) proof has been
given by A. Bellow, and her method has been extended to the case of compactly generated,
Abelian, locally compact groups of unitary operators by V.Graham [1974].
Several further characterizations are available for weak mixing: England and Martin
[1968] proved that an automorphism τ of a probability space is weakly mixing iff, for any
А, В of positive measure, the set of integers η ^ 0 with μ(Αητ~ηΒ)>0 has density 1.
Related results for Markov processes appear in Falkowitz [1973a]. Call a sequence (Bk)
remotely trivial if the intersection of the σ-algebras &n generated by {Bk, k7>n} contains
only nullsets and their complements. Krengel [1972] proved that τ is weakly mixing iff
every sequence (т~кА), (А e <stf), contains a remotely trivial subsequence, and that the
operator/->/° τ has discrete spectrum iff there exists no A with 0 < μ(Α) < 1 for which
(τ~* A) has a remotely trivial subsequence. (This is closely related to L. Sucheston's [1963]
characterization of mixing: A sequence (Ak, к ^ 0) is mixing (i.e., it satisfies
lim„ (μ(Αη η Ак) — μ(Αη) μ(ΑΗ)) = 0 for all к ^ 0) iff every subsequence contains a further
subsequence which is remotely trivial; see also Renyi [1963] and Jones [1972a]). A step in
Krengel's proof which has turned out to be useful is:
Theorem 3.11. If τ is an endomorphism in a probability space, and jtfuds the smallest σ-
algebra in which the eigenvectors of the operator Tf = f°xin£ = L2 are measurable, then
%uds consists exactly of the я/uds-measurable elements of L2, and T~1j^uds = £#uds>
Call A weakly independent if there exists a subsequence k1 < k2 < ... for which the sets
T~kiA are independent. It follows from the results of Krengel [1972] together with
Furstenberg's theorem 3.10 above that τ is weakly mixing iff the family of weakly
independent sets is dense in л/, (and also iff the family of weakly independent partitions is
dense in the family of all finite partitions of Ω). For a refinement of this, see Friedman
[1979].
If the σ-algebra s/ is countably generated and τ weakly mixing there exists a fixed
sequence McNof density 1 such that limneM μ (A m""Β) = μ(Α)μ(Β) holds for all
A,Bejtf. This has been observed by Jones [1972].
(Proof Let A1,A2,... be dense in s/. Find Mk of density 1 such that the limits exist
along Mk for A, Be {AUA2, ...,Ak}.lfnl <n2< ... increases fast enough the union Μ of
the sets [Пк-иnkLn(^in··-nMk) has density 1 and Mk contains {me M:m^ nk}).
The theory of weakly mixing flows largely parallels the theory for a single T, see Hopf
[1937] or Cornfeld-Fomin-Sinai [1982].
Let us mention some relatives of the notion of weak mixing: Furstenberg and

The splitting theorem of Jacobs- Deleeuw-Glicksberg 103
Weiss [1977] call τ mildly mixing if every fe L2, for which there exists пг<п2< ... with
у у ο τη* —/112 -► 0, is constant. They call such frigid. As every eigenfunction is rigid, mild
mixing implies weak mixing, τ is mildly mixing iff τ χ τ' is ergodic for all ergodic
automorphisms τ' of σ-finite measure spaces. The space spanned by the rigid functions/was first
considered by Foguel [1964]. A notion slightly stronger than mild mixing was defined by
Friedman and Ornstein [1971]. They called τ partially mixing if there exists β > 0 such
that А, Вея/ implies μ(Αητ~ηΒ)^ βμ(Α)μ(Β); see also Friedman [1984] and
Aaronson-Lin-Weiss [1979].
Beck [1960] generalized the notion of eigenvalue: A unitary operator Fona Hubert
space Ж is called eigenoperator for τ if there is a measurable/: Ω -> Ж with V(/(ω))
= /(τω) μ-a.e.. Flytzanis [1980], [1978] has studied rigidity and related concepts in this
context.
Natarajan and Viswanath [1967] called τ weakly stable if D-limμ(Α ητ~ηΒ) exists for
all A, B. It is shown that this holds iff 1 is the only eigenvalue. In the ergodic case weak
stability and weak mixing are equivalent. Kallenberg [1980] has investigated the
analogous concept for flows.
A comprehensive treatment of the recurrence theory initiated by Furstenberg [1977]
can be found in Furstenberg [1981], Furstenberg-Katznelson [1978], and Furstenberg-
Weiss [1978]. An introduction has been given by Furstenberg [1981a]. Furstenberg-
Katznelson-Ornstein [1982] have given an exposition of a streamlined proof of the
"ergodic Szemeredi theorem" (statement (3.6) for τ,- = τ1). Lesigne [1981] has proved
convergence for averages of type (3.6) in some cases. He treats the convergence in the Lr
n-l
mean of η'1 Σ giA^fi^i0^) ^orf,geL2, where the τ,- are commuting homeomor-
fc = 0
phisms of a compact metric space. For totally ergodic τ he also shows Lp-convergence of
η'1 "Σ (f° τ*) (g ° τ2*) (h ο τ3*) with/, g, h e Le.
k = 0
§ 2.4 The splitting theorem of Jacobs-Deleeuw-Glicksberg
A semigroup 5^ of continuous linear operators in a Banach space X is called
weakly almost periodic if for any χ e X the orbit if χ = {Tx: Те if) is
conditionally weakly compact. In this section we show that for Abelian weakly almost
periodic У the space X splits into a direct sum of the closed space Xuds spanned by
the eigenvectors with unimodular eigenvalues and the space Xfl of "flight
vectors", having 0 in their weak orbit closure. We discuss the non Abelian case in the
Notes.
We begin with some auxiliary results. If S is an abstract semigroup endowed
with some topology we say that multiplication in S is separately continuous if for
each s e S the maps t -+ st and / -* ts are continuous. The multiplication is said
to be jointly continuous if the map (s, t) -> st of S χ S into S is continuous. We
always assume that S contains a unit e, i.e., an element that satisfies et = te = t
for all t e S. A semitopological (topological) semigroup is a semigroup with unit
which is a Hausdorff space in which multiplication is separately (jointly)
continuous. A sub-semigroup / с S is called a left ideal if SJ= {s · /: se S,te J} is

104 Mean ergodic theory
contained in /, a right ideal ifJS cz J, and an ideal if it is a left and right ideal. A
minimal left ideal is a left ideal containing no other left ideal of S.
Theorem 4.1. Any compact Abelian semitopological semigroup S contains a
unique minimal ideal K, the kernel of S. К is contained in any ideal, and
(4.1) K=f)tS.
teS
К is a group. Ifq is the unit of this group, K— qS.
Proof If Ju..., Jk are ideals, their product Jx · J2 *... Jk is non empty, and is equal
to their intersection. Consequently, the family of closed ideals of S has the finite
intersection property. As S is compact the intersection К of all closed ideals is
non empty.
If /is any ideal, and t e /, then tS is a closed ideal contained in /. Hence, К is
the intersection of all ideals. As each /5* is an ideal and each ideal contains an ideal
of the form tS, (4.1) follows. Since AT is the intersection of a family of ideals, К is
an ideal. Clearly К is minimal and unique.
For any se K, sK is an ideal contained in К and, therefore, equal to К by the
minimality ofK. Therefore there exists q e AT with sq = s. By Ks — sK— К there
exists, for any / e K, an г e К with rs — t. Hence tq = rsq = rs — t shows that
tq = / holds for all / e K, and q is a unit. Finally. sK = К and q e К implies the
existence of an inverse of s in K. The last assertion follows from qs с KS с AT and
(4.1). d
Let & = wo-cl (Sf) denote the closure of if cz jSf(3E) in the weak operator
topology.
If {7^} is a generalized sequence and Ta, i?, Г belong to JSf (3E), then wo-lim Ta
= Τ implies (TaRxhhj} -* (TRxi9hfr and (RTaxh hj} = (Taxh R*hj} -*
<Txb R*hj} = (RTxh hj}. Therefore, multiplication in J5f (3£) is separately
continuous with respect to the weak operator topology. If Sf is a semigroup in if (X)
we have 5^5^ с Sf and thus ifS? cz <? by separate continuity. Applying separate
continuity once more and using wo-cl(S?) = S? we find S?S? cz SP, i.e., S? is a
semigroup. It is equally easy to check that S? is Abelian, if Sf is Abelian.
If Μ < oo is a bound for the norms of the operators in У we have | <Гх, A> |
^ M||x|| P|| for all Те Sf9 all xe X and all Ae £*. Therefore this inequality
holds also for all Те S? and Μ must be a bound for the norms of the operators in
We remark that multiplication in «Sf (X) in general is not jointly continuous
with respect to the weak operator topology. As an example let X be a Hubert
space spanned by a family {ei9 ie Z} of orthonormal vectors. If Γ is the unitary
operator determined by Tet = ei+1, (i e Z), and T0 the zero operator T0x = 0 we
have wo-lim Γ" = Γ0 and wo-lim T~n = T0, but wo-lim Tn · (Γ~η) = 7 φ T0 T0.

The splitting theorem of Jacobs-Deleeuw-Glicksberg 105
Lemma 4.2. A subset if a if (X) is relatively wo-compact iff ifx is relatively
weakly compact in Л for each xeX.In this case if is bounded and if χ — w-cl {ifx)
holds for all x.
Proof If if is relatively wo-compact, £f is wo-compact, and ifx is weakly
compact as the continuous image of a compact set. Conversely, assume that ifx is
relatively weakly compact for all x. Then ifx is norm bounded for all x, and by
the uniform boundedness principle if is bounded. For each χ let Ex be the weak
closure of ifx with the weak topology and Ε the Cartesian product of the spaces
Ex with the product topology. As each Ex is compact, Ε is compact by
TychonofFs theorem. Now let {Ta} be an arbitrary net in if. Then {{Тлх)хеХ} may
be regarded as a net in Ε and by the compactness of Ε there exists a subnet
{(ΤβΧ)Χ€Χ\ converging in E. This means that w-lim Τβχ exists for all x. Define an
operator ГЬу Tx — w-lim Τβχ. Clearly, Τ is linear. If Μ is a bound for the norms
of the operators in if we have || ГЦ ^ Μ as above. Thus Γ belongs to ^, and Τ
— wo-lim Τβ. As {Ta} was an arbitrary net we have proved that if is relatively wo-
compact.
In the last assertion the inclusion id follows from the weak compactness of
ifx, and the inclusion с from the definition of the weak operator topology, α
We remark that the lemma and its proof remain valid when the weak operator
topology is replaced by the strong operator topology and the weak topology in X
by the norm topology.
Definition 4.3. Let if с if (X) be a semigroup. A vector χ is called reversible if
for any Те if there exists an R e Ρ with RTx = x. A vector χ is called a flight
vector if there exists anSe^ with Sx = 0.
Let Xrev denote the set of reversible vectors and Xfl the set of flight vectors.
Clearly, Xrev η Xfl = {0}. We shall only be interested in the case when if is weakly
almost periodic. By the previous lemma this means that if is relatively wo-
compact, and the identity Ρ χ = w-cl {if x) then yields
%rev = iX e Х- У e W"°l i^X) => X e W"C1 {&У)} 5
Xfl ={xeX:0ew-cl(Sfx)}.
If if is an Abelian weakly almost periodic semigroup in if (X), if is an Abelian
compact semitopological semigroup with the weak operator topology and we
can apply theorem 4.1.
Theorem 4.4 (Jacobs-Deleeuw-Glicksberg). Let if be an Abelian weakly almost
periodic semigroup in if(X) and Q the unit in the kernel K= K(if) of P. Then
%rev — QX and Χ/ι — Q1 (0) = (I — Q) X- In particular, X is the direct sum of the
closed invariant subspaces Xrev and Xfl. The restriction of if to Xrev is a group.

106 Mean ergodic theory
Proof. If χ is reversible there exists Re if with RQx = x. Hence QRx = χ and
χ e QX. For the converse observe that for any Те if there exists an S e К with
STQ = Q, because AT is a group and TQ e K. If χ is of the form χ = Qy for some
>>, we therefore obtain ST χ = STg>> = £?y = x. This shows that all elements of
QX are reversible.
Obviously all elements of (2-1(Ό) = ix: Qx = 0} are flight vectors. On the
other hand, if χ is a flight vector, there exists an R with Rx = 0. Hence QRx = 0.
As QR belongs to К there exists S with SQR = Q. Thus, βχ = SQRx = 0.
(2-1(0) = (I— Q)X follows from the projection property Q2 = Q. As any χ
can be written in the form χ = Qx + (/ — (?) χ we find χ = xx + x2 with χ χ e Xreu9
x2 e Xfl. Conversely, if χ is of this form we have Qx = Qxx + 0 and xt = Qyx for
some yx. Hence Qx = Q2yx = Qy1 = xx. The splitting is therefore unique. The
group property of 9* in Xrev follows from the group property of Κ. α
A vector χ Φ 0 is called an eigenvector for if if there is a map λ: if -* С with
7x = Я(Г)х, (Гg 50. If Ш is a net and T= wo-lim 7; the numbers λ(Τα) must
converge to some λ (Τ). Therefore eigenvectors for if are eigenvectors for 9 and
conversely, χ is called an eigenvector with unimodular eigenvalues if |Я(Г)| = 1
holds for all Τ Let Xuds be the closure of the subspace of X spanned by all
eigenvectors with unimodular eigenvalues. We may say that if has unimodular
discrete spectrum in Xuds.
Theorem 4.5 (Jacobs-Deleeuw-Glicksberg). If if is an Abelian weakly almost
periodic semigroup in if (X), then Xrev = Xuds.
Proof It is easy to show the inclusion =5: For χ e Xuds and Те 9 the identity Tnx
= λ{Τ)ηχ shows the existence of a sequence щ with Tnix -* χ. It follows that, for
any hi9 ...,hmeX* and ε > 0, the set {Se tf: |<STx — x, ht}\ rg ε, i = 1,..., m}
is non empty, and any finite intersection of such sets is non empty. By the
compactness of 9 in the weak operator topology the intersection of all sets of this
type is non empty. Therefore there exists an S e 9 such that | <STx — x, h} | rg ε
holds for all ε > 0 and all h e £*. But then STx = x. As Те if was arbitrary χ is
reversible.
The proof of the opposite inclusion depends on some abstract harmonic
analysis and on the theorem of Ellis which asserts that any compact semitopological
group is a topological group; see Ellis [1957].
By this theorem К = K{if) with the weak operator topology is a compact
Abelian group. Let Г be its character group and let ρ denote the normalized Haar
measure on K.
For ye Γ let у (Г) be the complex conjugate of y(T). We now describe how
"weak integration" can be used to define operators Ty on X. First an element Tyx
of 3E** can be defined by

The splitting theorem of Jacobs-Deleeuw-Glicksberg 107
(4.2) (Tyx, h} = J <j(T) Tx, hy Q(dT).
К
If we show that Tyx belongs to the canonical embedding кХ of X in £**, then Tyx
can be identified with an element of X, and (4.2) will be the meaning of
(4.3) Tyx = \y(T)TxQ(dT).
к
As the map Τ -> у (Т) Tx is weakly continuous on K, the set {y(T) Tx: TeK}is
weakly compact. By the Krein-Shmulian theorem (see Dunford-Schwartz [1958:
434]) its closed convex hull со {γ(Τ) Tx: Те К} =-· A is weakly compact. Hence
к A is compact in the w*-topology of £**. Approximating the integrals in (4.2) by
Riemann sums we notice that Tyx belongs to the w*-closure of к A and therefore
to к A. It is now clear that (4.3) defines a linear operator in X. Its norm is bounded
by sup {|| ГЦ: Те К) < oo.
For R e Κ, χ e X, h e X* we find
{RTyx,h} = <Tyx,R*h} = $Q(T)Tx,R*h}Q(dT)
= l(y{T)RTx,hyQ{dT) = y{R)\(y{RT)RTx,hyQ(dRT)
= y(R)(Tyx,h}.
This, and a similar computation using RT— 77?, shows
(4.4) RTy = y(R)Ty = TyR, (ReK).
As Q is the unit in Κ, γ (Q) = 1 and QTy = Tr Applying (4.4) to SQ e К we obtain
STy = SQTy = y(SQ) Ty = y(S) Tr This means that TyX consists of eigenvectors
with unimodular eigenvalues λ{Τ) — у(Т).
To finish the proof it now remains to show that Xrev is contained in the space U
spanned by the spaces TyX, (у е Г). If h e X* vanishes on U, then
(4.5) inT)<Tx9h>Q{dT)
к
vanishes for all ye Г and all xeX. Assume <βχ, Α> φ 0 for some x. As
Τ -* <Γχ, h} is continuous on K, this implies J lF(Tx, h}g(dT) Φ 0 for some
neighbourhood Fof Q. As the characters form an orthonormal basis in L2 (Κ, ρ)
(see e.g. Dunford-Schwartz [1963:944]) there are linear combinations of
characters arbitrarily close to 1F. But then the integrals (4.5) cannot vanish for all у, а
contradition. We have proved {Qx, A> = 0 for all χ and for all h vanishing on
U. Thus QX = Xrev is contained in U. α
Remarks. If xy φ 0 is an eigenvector with unimodular eigenvalues
{λ(Τ): Те if}, then λ is continuous on 5^, and λ(ΤΞ)χλ = Г^Хд = Τλ(Ξ)χλ
= Я (Г) Я (S) χλ implies that Я is a character. Thus the unimodular eigenvalues are
exactly the characters у with Ty Φ 0. For any χ e X, h e £* we have

108 Mean ergodic theory
(Ty2x,h} = $Q(T)TTyx,h}Q(dT)
= !<y{T)y{T)T7x9h>e{dT) = <T7x9h>.
Therefore Ty is a projection onto the eigenspace corresponding to y.
If Я is a unimodular eigenvalue for 5^, then Τλ Φ 0 and Г/Аоф0 for some
h0. For all>>e£ one has (y,T*Tfhoy = <ГЛ7>,А0> = <А(Г>:ГЛ>>, А0> =
<^,Я(Г)ГА*А0>.Hence Г/A0isan eigenvector of^* = {Г*: Г е 5^} with
eigenvalue A* defined by А*(Г*) = Я (Г). Conversely, if A is an eigenvector of 5^*
with unimodular eigenvalue A*, then <>>, Я*(7;*Г2*)А> = (y,TfTf_K> =
0,71*А*(Г2*)А> = <^Я*(71*)Я*(72*)А> holds for all ^. Thus Я(Г):=Я*(Г*)
defines a character of 5^. It follows from <7^x,A> = §J(T)(Tx,h}dQ
= $I(T)(x,T*h}dg = |Ι(Γ)Α(Γ)<χ,Α>ί/ρ = <x, A> that ΓΛ Φ 0. Thus A is a
unimodular eigenvalue of Sf. The relation А*(Г*) = А(У) defines a 1-1-
correspondence of the unimodular eigenvalues of У and 5^*.
Proposition 4.6. 7/"^ w an Abelian weakly almost periodic group in JSf (3£), then
xe X is a flight vector iff <x, A> = 0 holds for all eigenvectors h of У* having
unimodular eigenvalues.
Proof If h is an eigenvector with unimodular eigenvalue A*, then | <Гх, А> | =
| <x, Г* A> | = | А* (Г*)<х, A> | = | <x, A> | is constant on £f. If χ is a flight vector 0
belongs to Px, and <x, A> = 0.
For any A and any/e £*, T^f is = 0 or an eigenvector of 5^* with I*. If χ is
orthogonal to all eigenvectors of ^* with unimodular A, then 0 = <x, 7^*/>
= <7^x,/> holds for all/. Hence T^x = 0. As A was arbitrary χ has no
component in Xuds and is a flight vector, α
To complete the generalization of the spectral mixing theorem we still have to
find an extension of the assertion D-lim <Гпх, у} = 0 for flight vectors x. This
will be done under a separability condition.
A linear functional L on the space of bounded complex valued functions g on У
is called an invariant mean ifg ^ 0 implies L (g) ^ 0, L (H) = 1, and U e У implies
L(g о U) = L(g) for all g. g о U is the (left) translate (g о U) (T) = g(UT); (we
need not distinguish right and left for Abelian £f).
Theorem 4.7 (Lin). Let У be an Abelian weakly almost periodic semigroup in
J5? (X) andxe X. Assume that the closed linear subspace X{x) ofX spanned by Ух
is separable. Then the following are equivalent:
(a) x e Xfl;
(b) there exists a sequence T{ e У with Txx -> 0 weakly;
(c) for every he X* and every invariant mean L we have L(g) = 0, where g(T)
— |<7X h}\. (We may write L(|<7x, h}\) = 0 for short).

The splitting theorem of Jacobs-Deleeuw-Glicksberg 109
Proof, (a) => (b): As S?x is weakly compact, the Krein-Shmulian theorem
shows that со {Sf x) is weakly compact and hence weakly sequentially compact by
the Eberlein-Shmulian theorem. By the Hahn-Banach theorem co(5^x) is also
sequentially compact in the weak topology of X{x) since the limits are in
co(5^x) с X(x). But the weak topology of a weakly compact subset of a
separable Banach space is metric; see theorem V.6.3 in Dunford-Schwartz [1958].
(b) => (c): The closed unit sphere В in £* is w*-compact. Define continuous
functions/· on В by f(h) = | <7Jx, A>|. Then/; -* 0 pointwise and the/· are
uniformly bounded by ||x||M where Μ = sup {11 ГЦ: Те £f) < oo. By Lebesgue's
theorem/ -> 0 weakly in C(B). By Mazur's theorem (Yosida [1974: 120]) there
η
exists, for any ε > 0, a convex combination Σ аг/£ of norm < ε. This means that
η i = 1
Σ аг I <7Jx, A> | < ε holds for A e B. Hence
£«,ЮТ*,Л>1<вМ
i = l
for all Τ e Sf, and using the invariance under translations by T{.
1(\<Τχ^}\) = Σ*Μ\<Τχ^}\) = Σ*Μ\(ΤΤίχ^}\)
= Σ(Σ^ί\<ΤΤίχ,Η}\)^^Μ.
As ε > 0 was arbitrary (c) follows for he В and hence for all A.
(c) => (a): If A is an eigenvector of 5^* with unimodular Я* we have | <7x, A> |
= 11* (Γ*) <x, A> | = | <x, A> | for all Τ Hence (c) implies that χ is orthogonal to
A. (a) now follows from proposition 4.6. а
Now consider the special case of a semigroup 5^ = {Г°, Г1,. · ·} generated by a
single power bounded operator Τ If Z/ is a Banach limit on ^ ({0,1,...}) we can
define an invariant mean on У by L(g) = 11 (#') with g' (k) = g(rfc). By theorem
3.4.1 the condition (c) in the previous theorem implies
j + n-l
limsupH-1 Σ |<Гх,А>| = 0? (Ae£*),
η j i = j
which is slightly stronger than D-lim (Tlx, A> = 0. However, a simple argument
will show that (b) implies a yet stronger condition: the convergence is even
uniform on B.
Proposition 4.8 (Jones-Lin) Let Τ be a power bounded linear operator in a Banach
space X and xeX. If there exists a sequence (w£) with Тщх -> 0 weakly, then
(4.6) lim (sup {TV"1 Σ \<T*x, A>|: he B}) = 0.
N-►00 i = 0
Proof First assume || ГЦ ^ 1. В is w*-compact. Af(h) *=f(T*h) defines a con-

110 Mean ergodic theory
traction on C(B). The function f(h) = |<x, h}| satisfies Anif(h)-+0, hence
Anif-+ 0 weakly in С (В) by the dominated convergence theorem. For every
(signed) measure μ e C(B)* with μ = Α* μ we have </, μ} = (Anif, μ} -* 0 and
hence <μ,/> = 0. The Hahn-Banach theorem now yields/e cl((7—^4) С(2?)).
But then
sup {TV"1 ς' |<^£-х?/г>|:/ге jB} = H7V"1 Σ^/lhO.
i = 0 i = 0
Ifsup{||rn||:« ^ 0} = M> 1, we introduce a new norm by |||x||| = sup||rnx||.
Then || л: || rg |||x||| rg Af||x|| and Г is a contraction for the new norm. For he X*
|P||| = sup{|<x?A>|:|||x|||^l}^||A||.
Hence В is contained in the unit ball of £* under the new norm and the
assertion follows from the contraction case, α
Proposition 4.9 (Jones-Lin). Let Τ be power bounded in a Banach space X and
χ e X, then (4.6) is equivalent to
(4.7) lim TV"1 Σ |<Гх,А>| = 0 VAe£*.
N-►00 i = 0
Proof. Clearly (4.6) implies (4.7). The argument for the converse is nearly identi-
N-l
cal to that of the previous proof: Now TV"1 Σ ^'/converges to 0 pointwise and
i = 0
N-l
weakly and μ = Α*μ implies </, μ} = <7V_1 Σ Λ*/9 μ} -* 0. α
i = 0
Notes
If ^ is allowed to be non Abelian, then Xrev and Xfl need no longer be linear subspaces of
X. Jacobs has given the following simple example: Let Sf be the semigroup {/, T, S} in R2
given by the matrices
r-G »)· s-G !)■
Then £rey = {(x1? x2): x2 = 0}, ЭЕЛ = {(xl9 x2Y *i = 0} u {(x1? x2)· *i + xi = 0}· For
the semigroup defined by the transposed matrices one obtains
*rev — \\X1> X2)'- X2 = 0/ ^ K-^Ij X2)' Xl = Χ2)·> Xfl = \\X1> X2)· Xl = 0}·
Yet, splitting theorems can be derived under suitable assumptions when the statement
Xrev = XUds is weakened. A finite dimensional ^-invariant subspace D of X is called a
unitary subspace of X if the restriction of Sf to D is contained in a bounded group of
operators on D. (D is a unitary subspace iff it is possible to choose an inner product on D so
that the restrictions of all operators of Sf to D are unitary). Let Xap be the smallest closed
subspace of X containing all unitary subspaces. The vectors in Xap are called almost
periodic. (This definition is equivalent to the more usual definition; see Maak [1950],
Jacobs [I960]). Clearly Xuds с Хар. In the Abelian case equality holds. We are interested in

The splitting theorem of Jacobs-Deleeuw-Glicksberg 111
the splitting property
(Sp) Xfl and Xrev are closed ^-invariant linear subspaces of X, X is their direct sum,
and Xrev = Xap.
To formulate the principal result of Deleeuw-Glicksberg [1961] we need some definitions:
For general weakly almost periodic Sf the kerne^AT^) is the intersection of all two sided
ideals of Sf. It is the unique minimal ideal of £f.
Cb(£f) is the space of bounded continuous functions on Sf, and Cx(£f) is the smallest
uniformly closed subalgebra closed under complex conjugation and containing the
constant functions and all/of the form/(r) = (Tx, h} with xeX, he £*.
An invariant mean on a closed subspace D oliCb(Sf) is an element mofD* with m (1) = 1,
/^ 0 => m(f) ^ 0, and invariant under left and right translations.
Theorem 4.10 (Deleeuw-Glicksberg). Let Sf be a weakly almost periodic semigroup of
linear operators in a Banach space X. Then each of the following conditions is necessary and
sufficient for the splitting property (Sp):
(a) C%(ff) has an invariant mean;
(b) K(£f) is a compact topological group;
(c) К(5?) contains a unique projection.
Apart from the Abelian case treated above and solved by Jacobs [1956] for reflexive X, the
main case of interest seems to be the case of contractions in spaces having suitable
convexity properties:
Theorem 4.11 (Jacobs-Deleeuw-Glicksberg) Assume that X is a strictly convex Banach
space for which X* is strictly convex. IfSf is a weakly almost periodic semigroup of
contractions in X, (Sp) holds.
(Jacobs [1957] gave the proof for X, X* reflexive and strictly convex, with one of them
uniformly convex).
The splitting theorem for groups of unitary operators in Hubert space has already been
studied by Maak [1954]. (For a similar result see Godement [1948: 64]) The case of
bounded groups in Hubert space was settled by Jacobs [1955].
Our presentation of the Abelian case borrows from an unpublished manuscript of Lin
[1974], which, in particular, contains proposition 4.6 and theorem 4.7. The book of
Jacobs [1960] contains different complete proofs of theorems 4.4 and 4.5 and shows that
the reflexivity assumption in the original paper of Jacobs [1956] was only needed to assure
the weak almost periodicity of Sf. Jacobs [1956] also gave a characterization of flight
vectors resembling theorem 4.7.
Proposition 4.8 and 4.9 appear in Jones-Lin [1976]. Jones and Lin [1980] have shown
that the sufficient condition w-lim Тщх = О for (4.6) and (4.7) (which clearly is also
necessary when X* is separable, and also if {Tnx} is weakly sequentially compact) is not
necessary in general. If {Tnx} is w* sequentially compact in X**, then (4.6) is equivalent to
each of the conditions
(4.8) 0 is a weak cluster point of {Tnx};
(4.9) <x, hy = 0 for all unimodular eigenvectors h ofT*;
(4.10) for every λ with \λ\ = 1, ЦЛГ1 £ ληΤηχ\\ -* 0.
о
For further related material, see section 8.1 and Nagel-Palm-Derndinger [1984]. Jamison
[1964] gave a direct proof of the following consequence of the Jacobs-Deleeuw-
Glicksberg theorems:

112 Mean ergodic theory
IfX is a Banach space, Те J£(X), and{Tx, T2x,...} conditionally compact in the norm
topology for each x, then { Tn x} converges in norm for all χ iff there exists no eigenvector with
eigenvalue Αφ 1, |A| = 1.
Gundel [1979] studied mixing properties of semigroups of endomorphisms and the
mutual implication of such properties in the Gaussian case. Zimmer [1976] has a general
structure theory for semigroups {тд, g e G}, G a locally compact group, including a
generalization of theorem 3.11. The books of Berglund-Hofman [1967], Berglund-Junghenn-
Milnes [1978], Burckel [1970] are references for the theory of semitopological
semigroups.
The papers of Dye [1965], Kuhne [1982] and Schmidt [1982] are further references
concerning weak mixing for semigroups.
We remark that the results discussed here also yield mean ergodic theorems of the type
discussed in §2.1. If У is weakly almost periodic, also &~ = coSf is weakly almost
periodic. The space Xap for the semigroup 9~ consists only of fixed points of Sf, and the
space Xfl of 6Г consists of vectors χ with 0 e coZfx. The results of this section, however,
are intrinsically deeper.

Chapter 3: Positive contractions in Lx
While mean ergodic theory has been developed for rather general spaces, point-
wise ergodic theory for operators mainly deals with positive contractions in Lx
and closely related topics. This subject has flourished because it unifies ergodic
theorems for measure preserving and nonsingular transformations with results
from the theory of Markov processes.
In this chapter we discuss the basic theory, including the Hopf decomposition,
the Chacon-Ornstein theorem, Brunei's lemma, and the existence of finite
invariant measures. Generalizations and refinements are defered to subsequent
chapters.
§3.1 The Hopf decomposition
Let us first describe some probabilistic notions motivating the study of
contractions in Lv Throughout, (Ω, jtf) will be a fixed measurable space.
A mapping Ρ: Ω χ sf -* R+9 (ω, A) -* Ρ(ω9 A)9 is called a substochastic kernel
if
(kl) for each A e s/, P(-, Α): Ω -* U+ is measurable, and
(k2) for each ω e Ω, Ρ{ω9 ·): sf -* R+ is a measure with Ρ{ω9 Ω) ^ 1.
Ρ(·9 ) is called a stochastic kernel, if even Ρ {ω, Ω) = 1 holds for all ω e Ω. We
may think of Ω as a set of "states" in which a system observed at time 0,1, 2,...
can be, and of Ρ (ω, A) as the probability of passing from state ω into the set A of
states in one time unit.
If τ: Ω -* Ω is a measurable transformation we may define a stochastic kernel
Ρτ(ω, A) = 1Α(τω). This kernel describes the deterministic transition ω -* τω.
Let b<stf denote the space of real valued bounded ^-measurable functions on
Ω, and let Μ — M{st) be the real Banach space of signed measures ν with finite
total variation || ν ||. A substochastic kernel induces important linear mappings in
these spaces by
(1.1) Τ*Η(ω) = $Η(η)Ρ(ω9(Ιη)9 (heb^)9
and
(1.2) Tv(A) = lP(<o9A)v(d<o)9 (veJ).
If h (η) is a reward one can receive in state r\9 then Γ* h (ω) may be interpreted as

114 Positive contractions in Lx
the expected reward after one transition given a start in ω. In particular Γ* 1A (ω)
= Ρ(ω, Α).
For an interpretation of Tone may think of a non negative ν as a distribution
of matter. The matter in ω is scattered at random according to the transition
probabilities. Tv is the new distribution of the matter. If ν assumes also negative
values, then v = v+ — v~.v~ may be thought of as antimatter. Some of the matter
and some of the antimatter may cancel each other after the transition. If ν{άώ) is
the probability that the system is in άω at time и, then Tv(A) is the probability
that the system is in A at time η + 1.
<v, h) = J hdv is a bilinear form on Μ χ hsi. The Fubini theorem for kernels
asserts that
J [J Η(η)Ρ(ω, άη)2 ν(άώ) = jAfo) [f Ρ(ω, Λ0 v(<fo)].
It follows that
(1.3) <7v, A> = <v, Г* A>, (veJJe iw*).
For the study of ergodic theorems we need also a reference measure. Let μ be a
σ-finite measure on (Ω, «я/), and
Lx (μ) = Lx = {veJiw <μ).
We say that a kernel Ρ is null preserving if μ (/1) = 0 implies Ρ (ω, yl) = О μ-a.e..
If τ is a measurable transformation, then the stochastic kernel Ρτ is null preserving
iff τ is null preserving. In this special case the operators defined by (1.1) and (1.2)
have the form h -> h ° τ, and v->v° τ-1.
Lemma 1.1. The kernel Ρ is null preserving iff Τ maps Lx into Lv
Proof. Assume Ρ null preserving, ve L1? and μ{Α) — 0. Then
TV (A) = IP (со, A) v(dco) = 0. Hence Tv <^ μ. Conversely, if Ρ is not null
preserving, there exist А, В with μ (A) = 0, 0 < μ (Β) < oo and Ρ (ω, Α) > 0 for ω e B.
Then v, defined by v(C) = μ{€ηΒ), belongs to Ll9 but 7V not. α
For many purposes the assumption that Ρ is null preserving is not a restriction of
generality. If μ is a given σ-finite measure and μ' a finite measure with μ' ~ μ, then
00
Ρ is null preserving for the measure μ = Σ 2~{Т{μ! > μ; (exercise). However,
i = 0
note that μ-integrable functions need not be /i-integrable.
It will be convenient to work mainly with Lx instead of Lv By the Radon-
Nikodym theorem there is a (linear, isometric, order preserving) isomorphism φμ
between Lx and Lx associating with each ν e L1 its Radon-Nikodym derivative
φμ(ν) = dv/άμ. If the kernel Ρ is null preserving, we may therefore regard Tas an
operator in Lx by identifying Lx and L±. More precisely: Tin Lx is the operator

The Hopf decomposition 115
Tf— φμΤφ~χ/. Because of the identification we use the same symbol for the
operators in Lx and Lv Γ is a positive contraction in Lv
If the kernel Ρ is null preserving, (1.1) defines also an operator Γ* in /^(μ)
= L^: In this case μ(Α) = 0 implies Γ* 1A = 0 μ-a.e. and, by approximation with
step functions, h = 0 μ-a.e. implies T*h = 0 μ-a.e.. L^ is the dual space of Ll9
with scalar product </, A> = J/· Λί/μ, and, by the identification of Lx and L1?
(1.3) translates into <Γ/, A> = </, Γ*Α>. Γ* is the adjoint of T, and the notation
is justified.
One may ask if each positive contraction Tin Lx is given by a null preserving
substochastic kernel. This is in fact true, if (Ω, si) satisfies some regularity
conditions, e.g., if Ω is a complete separable metric space and si the σ-algebra
of Borel sets; see e.g. Neveu [1965: 192]. In the general case we may still
find a "generalized" null preserving substochastic kernel: For fixed A,
Ρ (ω, A) := Γ* 1A (ω) defines Ρ(·,Α) only up to nullsets. It is not hard to see that
one may choose Ρ (ω, A) in such a way that (kl) and the null preserving property
hold, and (k2) is replaced by
(k2i) for all ω β Ω, A e si, О й Ρ(ω, А) й 1, and
00 00
(k2ii) for all disjoint Al9 A2,... e si, Ρ{ω, (J At) = Σ Ρ(ω> ^д μ-a.e..
1 i=l
If the equation in (k2ii) would hold everywhere, P( ·, ·) would be a null preserving
substochastic kernel.
We shall develop the theory for positive contractions in Lx rather than for
kernels, although the gain in generality is small. The advantage consists mainly in
greater elegance and shortness of certain arguments.
The equations (1.1) and (1.2) still make sense for the generalized kernel P{ ·, ·),
h e L^, and ν e Lx. But P(co, ·) need not be σ-additive, and the integral in (1.1)
must be defined as the limit of
\hMP(',dn)=k±aUnP{',AUn)
i = l
where hn = Σ αι,η ^AUn is a sequence of step functions with \\hn — hЦ^ -* 0.
i
A positive contraction 5* in L^ is called a sub-Markovian operator if Shn -> 0
a.e. holds for any sequence hn decreasing to 0. If Tis a positive contraction in Lu
it is easy to see that T* is a sub-Markovian operator. Conversely, if S is a sub-
Markovian operator, we may define a positive contraction Tin Lx with T* = S
by Tv(A) = \S\Ad\, Tand Г* are called Markovian if Г*1 = 1. An equivalent
condition is J Τ/άμ = \/άμ for all/e Lx. (Simply use \/άμ = </, 1>.)
Sometimes it will be useful to extend the range of definition of T: For
measurable/with/" g Lx we may uniquely define Tf= lim Tfn where fn is an increasing
sequence of integrable functions tending to/; see § 1.6. Similarly, one may extend
the range of definition of T*.
We have set the stage and may start the investigation of positive contractions in

116 Positive contractions in L1
Lu or - from the dual point of view - of sub-Markovian operators. We first give
three different but equivalent descriptions of the important Hopf decomposition
of Ω into two sets С and D — Ω \ C, called the conservative and the dissipative part
of Ω (for T).
Definition 1.2. A function h with T*h = h is called harmonic, a non negative
function h with h ^ Γ*Α is called super harmonic. A superharmonic A is called
strictly superharmonic on A if h > T* h on A.
(We may delete "a.e." since we are dealing with equivalence classes of functions.)
Theorem 1.3. If Τ is a positive contraction in Lu there exists a decomposition of Ω
into disjoint sets C, Z>, determined uniquely mod μ by:
(CI) Ifh is superharmonic, then h — T* h on C\
(Dl) There exists a bounded superharmonic A0, which is strictly superharmonic
on D.
h0 may be constructed with the additional properties T*nh0 -> 0 on D, and h0
= 0 on С
Proof. Let us say that a set A has the property (ss) if there exists a bounded
superharmonic gA with gA > T*gA on A. By an exhaustion argument (see § 1.3)
we may find countably many Di9 D2,... with the property (ss), such that no
00
subset of positive measure of С = Ω\Ό with D — (J Dt has the property (ss). If
oo i = l
ct > 0 is a bound for gDi, then Ax = Σ ^χ2~^Όί is bounded, superharmonic,
i = l
and strictly superharmonic on D. Assume that there exists a set В cz С with
μ (Β) > 0 and an unbounded superharmonic hB with hB > T*hB on B. Choose
some α for which B' = Bn{hB 7> oc} η {Τ*hB < a} has positive measure. Then,
because of 0 ^ Γ*1 ^ H, the function A = hB л α is bounded, superharmonic,
and strictly superharmonic on В'. This proves (CI), (Dl) and the uniqueness.
By induction the sequence hn = T*{n~1)h1is decreasing and non negative, and
each hn is superharmonic. Put hao = limhn. As Г* is sub-Markovian T*hO0
= lim Г*А„ = lim An+1 = A^. We may take h0 = ht — h^. α
Corollary 1.4. Г*1=Ь«С.
n-l
Recall the notation Snf= Σ Tlf («eNu{oo}).
i = 0
Theorem 1.5. The sets C, D of the Hopf decomposition are also determined
uniquely mod μ by:
(C2) ForaIlheL+9S*h = ao on Cn{S*A>0} and
(D2) there exists an hD e L J with {hD > 0} = D and S% hD rg 11.

The Hopf decomposition 117
Proof. Multiplying A0 by a positive constant we may assume A0 rg 11. hD — h0
— T*h0 is strictly positive on D and vanishes on C, and S*hD = (A0 — T*h0)
+ (T*h0 - Γ*2Α0) + ... + (T*{n~l)h0 - T*nh0) й h0. This proves (D2). Now
assume the existence of an A e L*, an η e N9 and an А с С with μ(/1) > О,
Γ*ηΑ > 0 on ,4 and 5* A < oo on A. Consider A = Min (1, Σ Г*1'/?). Г*И ^ И
со i = n
andr*A^ Σ Γ*' A imply Γ* Α ^ A?and Г*0+1)А ^ Г^'А for ally ^ 0. Hence
i = n + l
00
r*^'A = A on С for ally. But T*jh ^ Σ Γ*1'Λ tends to 0 on A. This contradicts
~ i=n+j
to h>0 on A, and proves (C2). D
Using theorem 1.5 it is not hard to check that the conservative part studied
in §1.3 for a null preserving transformation τ coincides with the conservative
part for the contraction ν -* Tv = ν ο τ"1, or, equivalently, for Γ* given by Г* А
= A ° τ. Extending the terminology used there we may call Γ and T* conservative
if Ω = С and dissipative if Ω = Z>.
The Hopf decomposition generalizes also another well known decomposition.
Consider a Markov chain on an at most countable state space Ω given by a matrix
Ρ = (pik) of transition probabilities (/, к e Ω). Let μ be the counting measure:
μ (A) = card (A). Then P(i9 A) — Σ;€ aPij *s a stochastic kernel, which trivially is
null preserving, because there are no non empty nullsets. If Pn = (p"k) is the
matrix of the «-step transition probabilities, the contraction in L^ belonging to
the kernel may be described by Γ*η l{fc} (i) = pnik. Recall that a state i e Ω is called
00 00
recurrent if Σ Pu = °° and transient if Σ Pu < °° · Theorem 1.5 implies that С is
n=l n=l
the set of recurrent states and D is the set of transient states.
Finally, we describe the Hopf decomposition in terms of T:
Theorem 1.6. The sets C, D of the Hopf decomposition are also determined
uniquely mod μ by:
(C3) For allfe L+u £«,/ = oo on С η {5'00/> 0}, and
(D3) for allfe L\9 5'00/< oo on D.
Proof <£,/, AD> = </, S* AD> ^ </, 1 > < oo and hD > 0 on D imply (D3). Now
assume Saof< oo on a set fcCn{/>0} with 0<μ(70<οο.
A = 1F(1 + S^/)-1 is strictly positive on F. By (C2) S*A = oo on F. But
</, 5* A> = {S^f A> = J(^/)lFa + Ζ^Γ'άμ й μ(Ρ) < oo,
which contradicts to/> 0 on F. Hence 5^/= oo on С η {/> 0}. Repeating the
argument with Tnf we find that 5^/= oo on Cn{Tnf> 0} for all л ^ 0. α
We shall use the notation Lp(B) = {fe Lp: {/Ф0}с 5}, (1йрй oo).

118 Positive contractions in Lx
Definition 1.7. A setBestf is called T-absorbing if TfeL1(B) holds for all
feLx(B).
Intuitively: if/is a distribution of matter with support in В then Г cannot carry
any of the matter into Bc. The probability of a transition from a state in В into Bc
is zero.
Theorem 1.8. The conservative part С for a positive contraction Τ in Lx is T-
absorbing.
Proof. Let ρ be a strictly positive element of Lx. By (C3) and (D3), С = {S^p
= oo}. For/e L\ (C) we have nfu S^p for all n. Hence nTf£ TS^p = £ 7>
i = l
for all л. In particular, S^p — oo on {7/> 0}. This proves 7/e LX(C). α
Sometimes the following characterization of Γ-absorbing sets in terms of Γ* is
useful:
Theorem 1.9. Be si is T-absorbing iff Г*lBe й V
Proof First let В be Γ-absorbing and let/be an element of L\ with {/> 0} = B.
Then Tfe LX(B) implies 0 = <Jf 1BC} = </ Г*1вс>. Hence Г*1вс = 0 on B.
Using Г*1Вс ^ H we obtain T*1BC й 1вс. On the other hand, if this inequality
holds and g belongs to L\ (Я), then 0 й <Tg, lBc} = <g, Г* 1BC> ^ <g, lBc> = 0
and Tg must belong to Lx (5). For general g e Lx (B) use g = g+ — g". D
Notes
The foundations of the ergodic theory of positive contractions in Lt were developed by
E. Hopf [1954]. Foguel [1979] noticed that one does not need the maximal ergodic
theorem. The theory of positive contractions in Lx has been discussed in the books
of Neveu [1965], Foguel [1969], Garsia [1970], and Revuz [1975].
Mutatis mutandis, most of the results of § 1.3 admit generalizations to positive
contractions in Lv For a positive contraction Г in Lx the conservative part agrees with that of Tk
for all к e N. If {Tt, t > 0} is a strongly continuous semigroup of positive contractions, all
00
Tt have the same conservative part, and it agrees with { J TJds = 00} for all strictly
0
00 00
positive fe Lv If φ(ή is decreasing, ^ 0, with J φ(ή = 1 and J (p(t)tdt < 00, also the
о о
00
operator Q with Qg = J φ(ή Ttgdt has the same conservative part; see Horowitz [1974].
0
Hoover, Lambert and Quinn [1982] have determined the conservative part for operators
Tf=hfoT.
An extensive bibliography on Markov chains with a general state space has been
prepared by §idak [1976].

The Chacon-Ornstein theorem 119
The conservative part of a Markov shift. Let (E, J% ρ) be a σ-finite measure space and Га
positive contraction in Lx (ρ) with conservative part C, and with T* 1 = 1. As in § 1.4 take
Ω+ = Ez+, Xj =j-th coordinate map. Generalizing 1.(4.6) put
b(X0eA09...,XneAJ = llAoT*tAl...T*lAHdQ.
Under slight assumptions (e.g. Ε is a Borel set in a Polish space, and !F its Borel subsets),
μρ extends to a measure in the product σ-algebra. The shift θ is null preserving in (Ω+, μρ).
In the case ρ = Τρ the shift is /^.-preserving and we can consider the bilateral extension to
Ω = ΕΖ.
Theorem 1.10. The conservative part of θ in Ω+ isCz+, and the conservative part of the shift
in Ω is Cz.
This was proved by Harris and Robbins [1953] for Ω and countable E, and by Moy [1965]
for Markov processes with transition probabilities having a density. The general case, due
to Simons [1971], is similar.
When X0, Xl9... is an aperiodic, transient, irreducible random walk on Ε = Ζ, it may
be seen that the remote σ-algebra л/т is trivial. Hence, Θ is then an ergodic dissipative
measure preserving transformation; see Krengel-Sucheston [1969]. [In Ω + it is clear that
the ^-invariant sets belong to s/^. Krengel-Sucheston showed that this remains so for the
bilateral shift in the conservative case, but not in the dissipative case.]
Several authors have considered the problem of approximation of certain Markov
operators by operators with additional properties or by convex combinations of such
operators; see Brown [1966], Kim [1972], Iwanik [1980], Choksi and Prasad [1983].
An interesting class of Markov operators is the class of convolutions by a probability
measure ρ on a locally compact group G. A stochastic kernel is given by Ρ(ω, A)
= ρ(Αω_1). Then ρ is the distribution of the size of the steps in a random walk on G.
For simplicity let us assume that G is Abelian with Haar measure μ. The
operator Τ is given by Tv = μ * v. Its restriction to Εί(μ) has the form Tf(w) = μ */(ω)
= ]"/(ω — η)ρ(άη). Τ* is computed as T*g((o) = §g(co + η)ρ(άη).
We refer to the papers of Kerstan-Matthes [1965], Foguel [1975], Derriennic [1976],
Glasner [1976a], Lin [1977], and Derrienic-Lin [1984], for the study of the asymptotic
properties of these convolutions.
For the construction of conservative and dissipative Bernoulli-shifts with nonidentical
factor measures, see Krengel [1970], Hamachi [1981], Grewe [1983].
§ 3.2 The Chacon-Ornstein theorem
Let ГЬе a positive contraction in Lx. We shall now study the a.e.-convergence of
Snf/Sng on {S^g > 0} for/e Ll9 ge L\. This will be independent of the Hopf
decomposition.
It is of interest that part of the proof works for an arbitrary positive linear
operator Tin a general vector lattice of measurable functions on Ω (or of
equivalence classes of functions). For the heuristic interpretation we think of L = Lx
and of the transport of matter by T.

120 Positive contractions in Lx
We write/-^ g if/e L+ and there exist r,se L+ with/ = r + s and g = r -\- Ts.
(g results if the part r of the matter/remains fixed and the part s is mapped with
T). For «^2we write/A g if there exists a gx with/-^^* gx and gx -^ g. The
nonlinear operator U in L given by £/A = 77г+ — A" is called the filling operator. If
h = /— g with/, ge L+ then/may be considered as matter, g as antimatter.
U(f— g) is obtained as follows: First the part/л g of/and g cancel each other,
and then the remaining part h+ of /is mapped, while the remaining part h~ of g
stays where it is. Frequently, g is thought of as a hole. The part/л g of/falls into
the hole and partly fills it. h~ is the remaining hole. The part h+ of/which did not
drop into the hole is mapped. If we apply U a second time part of this mapped
matter Th+ will drop into the remaining hole and again fill it partly, the rest will
be mapped again. The repeated application of U is therefore called the filling
scheme.
Lemma 2.1. For any figeL and и ^ 0 there exists an fn e L+ with f-^fn and
Un{f-g)=fn-g.
Proof. For « = 0we may take/0 =/ If/„ has already been constructed, then
tn=fn-lUn(f-g)T is non negative and [Un{f-g)Y = -Un(f-g) +
LUn(f-g)T=g- tn. Hence Un+1 (/-g) = T(fn- tn) -(g- tn) and we may
put/n+1 = r(/n -/„) + /„. α
Intuitively /„ is the part of the hole that has been filled up to time η and/„ is the
sum of /„ and the remaining matter. Recall the notation Ml, Mn from § 1.2. The
next theorem asserts that within the set {M%h > 0} the hole will be filled
completely by time η — 1:
Theorem2.2. For heL, n^i, and T, U as above we have £/n-1A^0 in
{Mlh>0} = {Mnh>0}.
Proof. By the definition of U the sequence (Uk A)", (k = 0,1,2,...), is decreasing.
It is therefore sufficient to show that for almost all ω e {M%h > 0} there exists
some m with 0 rg m rg η — 1 and (Umh)+ (ω) > 0. This is equivalent to showing
к
that φλ(ω) = Σ (Umh)+ (ω) is strictly positive for some 0 rg Λ; rg и — 1. As
m = 0
(Umh)+ is that part of Tmh+ which did not drop into the hole up to time m one
should have
(2.1) q>k-l· Σ Tmh = Sk+1h.
m = 0
We can complete the proof by showing that (2.1) indeed holds. For к = 0 (2.1)
is the obvious inequality h+ ^ h. If (2.1) has been verified for some k, we obtain

The Chacon-Ornstein theorem 121
Sk+2f=h + TSk+if£h + T<pk = h+ Σ T(Umh)+
m = 0
= h+ Σ (Um+ih-(Umhy)
m = 0
= h+ Σ l(Um+1h)+-(Um+1hy + (Umh)-]
m = 0
= h + (<pk+1-h+) + h--(Uk+1hyu(pk+1· □
From now on Τ will be a positive contraction in L — Lv
Before we continue it may be useful to point out that theorem 2.2 yields a simple
proof of Hopf's maximal ergodic theorem: For A e Lx put/= h+,g = /Γ and find
/„_! with (fn-i — g) = Un~1h and f1*^-* fn-t as in Lemma 2.1. By induction
/**/' implies J/^ \f. As {h+ > 0} is contained in E-= {Mnh > 0} we obtain
Ε Ε Ε Ω Ε Ω ΕΕ
This is the maximal ergodic theorem with {Mnh ^ 0} replaced by {Mnh > 0}.
The slightly sharper formulation in theorem 1.2.1 is obtained by approximation:
If hk is a sequence in Lx strictly decreasing to h e Ll9 then the sets {Mnhk > 0}
decrease to {Mnh ^ 0} and therefore
J Ιιάμ = lim J Акф^0.
{Mnh^0} к {Mnhk>0}
The following lemma will be rather crucial:
Lemma 2.3 (Chacon-Ornstein). If Τ is a positive contraction in Lx,fe Lx and
geL\9 then lim,,^ Tnf/Sn+1g = 0 a.e. on {S^g > 0}.
Proof. We may assume /^ 0. For ε > 0 set rn = Tnf— sSn+1g and
Λ = {Tnf/Sn+1g> s}n{g>0} = {rn >0}n{g> 0}, (n ^ 0). We first give а
heuristic argument: Imagine that we are colouring the amount \Anzg of the
matter TnfdX time n. As the matter already coloured at time к < η has moved η — к
times, the amount of matter still available for the colouring at time η is at least
(Tnf— sTSng)+ ^ 1 Ansg and suffices. As the total amount of matter coloured in
all steps cannot exceed \/άμ we obtain
(2.2) J Σ 1^6g<//i£j/<//i<oo.
n = 0 Ω
Formally, rn = Trn^ — eg rg Tr+-X — eg, (n ^ 1). From this we derive r„ rg Tr*-X
— slAng by verifying this inequality on each of the sets {g = 0}, An9 {g > 0} η Acn.
Hence jel^g" ^ J Tr,^ — J rn+ ^ Jrn+_x — j"rn+. Summing over η ^ 1 and no tic-

122 Positive contractions in Lx
ing r£ = /— s lAog we indeed obtain (2.2). Thus, a.e. ω e {g > 0} belongs to only
finitely many An. As ε > 0 was arbitrary the convergence is proved on {g > 0}.
For the convergence on {Tmg > 0} repeat the argument with Tm/and Tmg. α
Lemma 2.4. Assume f9ge L\, и ^ 1, g1^ gl9 and y>\, then
{lim sup Sn(f~ g)>0}cz {lim sup S„(y/- gl) > 0}.
Proof. As у may be written as a product 7i · y2 · · · Ул with y£ > 1 we may assume
η = 1 (and then use induction). Now g1 — r + Ts with r + s = g and r,se L\. The
n-l
result follows from Sn(yf- gx) = Σ ^Чу/- г - 7ϊ) = 5и(у/- g) + s - Tns
^ $,(/- g) + (y - 1) Sn/- Г""1 (7У), because by Lemma 2.3 Γ""1 (7ϊ)
^(y-l)5B/forlargenon{15o/>0}. d
For Hesi.fe L\ and «^Owe define
Wif=sup{igdp.p>g} and Уд/= lim WnHf.
Η η->οο
The sequence f^/is increasing because/-^ g implies/3^1-* g. Heuristically, Ψ^/
is the maximum amount of matter we can bring into Я by mapping various parts
of/at most л times. As/-1* gx implies || gx\\ 1 ^ ||/||1? it is clear that ΨΗ/^\\/\\ν
Obviously Ψ % and ΨΗ are positively homogeneous: for α > 0, Ψ^{α/) = αΨ^/
and ΨΗ(μ/) = αΨΗ/. Let E^ (h) denote the union of the sets En(h) = {Mnh > 0}.
Lemma 2.5. For f,geL\, h=f—g and и ^ 1, Η cz En{h) implies Ψη1/
^ J 10£έ/μ and Η cz E^ (h) implies ¥Ή/ = J 1н«?Ф·
Proof. By theorem 2.2, Un~1h is non negative on En{h), and by Lemma 2.1,
Un-lh=fn^-g with f^^fn-v If Яс£„(А), then O^JIhI/-"1*
= J 1h(/w-i — g) implies the first assertion, and the second follows by applying
the first to Hn = HnEn(h) and letting η tend to oo. α
Lemma 2.6. Η cz {lim sup Sn(f— g) > 0} with fige L\ implies ΨΗ/^ ^hS-
Proof. If, for some η ^ 1, g-^ gl9 then by Lemma 2.4 lim sup Sn(yf— g^) is
strictly positive on Η for all у > 1. Hence Hcz Eao(yf— gj. By the previous lemma
γΨΗ/= ΨΗ(γ/) ^ JlHgi· As у > 1 was arbitrary ΨΗ/^ J 1Η£ι· As this holds
for all л and for all gx with g-^ gx the assertion follows, α
With these tools it is easy to prove the theorem of Chacon-Ornstein [I960]:
Theorem, 2.7. Let Τ be a positive contraction in Lu feLx and geUu then
Snf/Sng converges a.e. on {So0g> 0} to a finite limit.

Brunei's lemma and the identification of the limit 123
Proof, We may assume /^ 0. First notice that WHg is strictly positive for
any Hc{Saog>0} with μ(#)>0 because g^Tng. On {5'oog>0} we
set Ъ = lim sup (Snf/Sng) and h = lim inf (Snf/Sng). We need not define Ъ and h
elsewhere. On (h > a} we have lim sup (Sn(f— ag)/Sng) > 0. As Sng increases
this shows that (h > a} cz {lim sup Sn(f— ocg) > 0}.
Assume that Η = (h = oo} η {S^ g > 0} has positive measure. Then, by lemma
2.6, we obtain for all ос > 0
Ψη/ ^ Ψ η (*g) = *4>Hg with VHg > 0.
This is a contradiction to WHfu 11/111 < oo. Hence A < oo on {SOog > 0}. If
the assertion of the theorem is wrong there exists a subset С? с {S^g > 0} with
μ(<7) > 0 and some rational numbers a, /? with G a {h <a< /? <Ъ}. Now C? is
contained in {lim sup Sn(f— Д?) > 0} and in {lim sup Sn(ag — /) > 0}. By
lemma 2.6, ^/^ j»^^ and WGfu ^gS- This contradicts 0 < WGg < oo. α
Notes
The Chacon-Ornstein theorem unifies many results. The special case where Tis given by a
Markov chain with countable state space and/, g are indicator functions of single states
goes back to Doeblin [1938]. The special case where Tis given by a conservative nonsin-
gular transformation is due to Hurewicz [1944]; see also Halmos [1946]. For a Markoff
process with finite invariant measure early results were due to Doob [1938], [1948] and to
Kakutani [1940]. Here we have followed the approach of Akcoglu-Chacon [1970a]. It is
rather straight-forward to derive the continuous parameter variant of the Chacon-
Ornstein theorem from the present discrete time version.
Fukushima [1974] developed a potential theory for "standard Markov processes", and
he strengthened the Chacon-Ornstein theorem for this setting, showing convergence
except on an almost polar set. For generalizations; see Sur [1977], [1978], Fukushima
[1983]. Direev [1981] extended this to additive functionals.
§ 3.3 Brunei's lemma and the identification of the limit
We have not yet identified the limit of the sequence Snf/Sng in the Chacon-
Ornstein theorem. This is now usually done via a deep lemma of Brunei [1963],
which has considerably influenced the whole subject including the previous
section. The identification of the limit will lead to an ergodic theorem of Dunford-
Schwartz concerning Cesaro averages, and of a generalization due to Chacon.
We also discuss the question of convergence of Anf'm L^norm.
1. Equilibrium potentials. Let Γ be a positive contraction in Lx and α a
measurable function on Ω with 0 ^ α ^ 1. Then/ -► Taf*= α/+ Γ((1 - a)/) is a positive
contraction in Ll9 called a partial application of T.

124 Positive contractions in L1
The filling operator [/has the form Uf— 7^/with α = l{/<0}. For/, ge L\ the
relation f1* g is equivalent to the existence of an α with g = Τ J. Hence, for
Ее si,
Ε
where the sup ranges over all sequences consisting of η measurable functions
at Ω-+[0,1].
We are particularly interested in the case α = \E. If IE is the operator/-» IEf
= 1£/ then TE := T1e may be written as TE = /£ + 7У£с. The next lemma confirms
the intuitive feeling that this partial application of Г should be most efficient for
bringing matter into E:
Lemma 3.1. For allfe L\ andn^i, ΨΕ/= fE ΤΕ/άμ.
Proof. The lemma follows by induction if we prove that for all η ^ 0,/e L\, and
measurable α: Ω -* [0,1]:
(3.1) \TEnTJulTt'f.
Ε Ε
To this end set β = al£. We first show by induction that
(3.2) T£Ttf- T£TJ= (TIEC)"+1 (α/) - /£,(77£,)"(αΛ·
Indeed, for η = 0,
V- ^= /*/+ ПО - β)Ω - *f- τ((ί - a)/) = т{\Есф - \EC*f,
and if (3.2) holds for some n, then
7Г1 Tpf- Tt1 TJ= (IE + TIEC) ((77£,)n+1 (ofl - IEC{TIEcf{af))
= (77£,)"+2(αΛ + /£(77£,)"+1(«/) - (TI£c)/£<) (TIEC)n(af)
= (Γ/£,)η+2(α/)-/£,(Γ/£,)η+1(«Λ·
The identity (3.2) implies 1£ Γ£" Tju h П T,f.
We may therefore assume al£c = 0 in the proof of (3.1), and obtain
successively
TEf- TJ= (1£ - α)/- Γ((1£ - a)/)
Tif- TE TJ= (1£ - a)/- TE Г((1£ - a)/)
7Г7- Γ£"Γα/= (1£-α)/- Γ£"Γ((1£-α)Λ·
(3.1) now follows from
I TEn T(f(iE - a)) й f /(1* - «) = i/(l£ - a). π

Brunei's lemma and the identification of the limit 125
By induction we obtain
(3.3) τ£ = "ΣΐΕ(τιΕ<γ + (τιΕ<γ, (»^ΐ).
v = 0
Hence, for/e Uu
(3.4) У|/= J Σ (ΤΙΕ.γ/άμ = \№άμ
Ε ν = 0
and
(3.5) ΨΕ/= lim WSf= J Σ №<)ν/Φ = ί/ν£Φ
и Ε ν = 0
with
(3.6) νί·= Σ V*T*yiE and %:=lim%n= £ (4<Γ*)4£.
ν=0 η ν=0
(We need not distinguish IE and /f because also /£ has the form h -* 1£ A).
Inductively, we can see that т/>£ rg 11, because then Τ*ψΕ ^ H implies H ^ 1£ ν Τ*ψηΕ
= 1E + 1£сГ*тр£ = t/>£+1. Lemma 3.1 and lemma 2.6 now almost instantly yield
Theorem 3.2 (Brunei's lemma). Let Τ be a positive contraction in Lufe Llf and
Ε cz {ω: Snf(pS) > 0 for infinitely many n}, then \/\ρΕάμ ^ 0.
Proof. Let peLX be strictly positive in Ω and fe = f+ sp.
Then, for all ε > 0, £"is contained in {lim sup„_^00 Snf > 0}. Lemma 2.6 implies
ψε/Ϊ ^ ψε17'· When ε decreases to 0,/ε+ [f+ andy^T ]f~ a.e. and in Lrnorm.
As ψΕ is bounded,
\ΓψΕάμ = lim/Λ" Ve^ = Urn УЕ /?^ lim 5^/Г
ε ε ε
= lim J/~ ψ£^μ = J/" ψΕόμ>· °
ε
Heuristically, Brunei's lemma says that if £" is contained in {ω: Snf{(n) > 0 for
infinitely many n) then we can bring more mass into Ε starting with a mass
distribution /+ than with f~. The function ψΕ has been called the equilibrium
potential or the Brunei function of Ε in the literature. It is the minimal superhar-
monic function dominating 1£. In fact, the recursive relation between ψΕ and
ψΕ+1 is a special case of a construction in potential theory: For any g0 e L* put
8n+i —Sov T*gn, (η ^ 0). By induction (gn) is monotonely increasing and
ll&illoo й llgolloo· Ifgoo:=limgw, then Г*^ = lim7*g„ й limgn+1 = ^, as Г*
is sub-Markovian. If A is any superharmonic function with h ^ g then by
induction h^gn for all n. Thus, g^ is the minimal superharmonic function ^ g0.
It may also be worth mentioning the identity
(3.7) ψΕ = Τ*»1Ε

126 Positive contractions in Lx
which is trivial for η = 0 and follows for η ^ 1 from
If Γ is given by transition probabilities Ρ (ω, A) as in § 3.1, then the transition
probabilities ΡΕ(ω9 A) for 7^ are = Ρ (ω, A) for ω e Ec, and for ω e Ε, ΡΕ(ω9 ·)
admits only the deterministic transition ω -* ω. Γ/η1£(ω) is the probability of
being in Ε at time л given a start in ω subject to the transition probabilities
PE (·, ·). This is equal to the probability of being in Ε before time η or at time η
subject to the original transition probabilities. We may therefore interpret \ρΕ(ω)
as the probability of ever being in Ε given a start in ω.
2. Identification of the limit. Our next aim is to determine the limit of Snf/Sng in
the Chacon-Ornstein theorem on the conservative part C.
Let if be the family of Γ-absorbing subsets of C. First assume Ω = С. Then
T*1 = l is harmonic. Be si is Γ-absorbing iff \BC is superharmonic. As Τ is
conservative any superharmonic function is harmonic. But then 1B = l — 1BC
must be harmonic, too, and we see that В belongs to if iff Г* 1B = \B. Hence if is
an algebra, and even a σ-algebra because Г* is sub-Markovian.
In the general case if is a σ-algebra in С because the restriction of Τ to
LX(C9 Cns/, μ) is conservative and А с С is ^-absorbing iff it is absorbing
under this restriction.
Г is called ergodic ifT*lA = lA implies A = 0 or A = Ω mod μ. In the
conservative case this means that if is trivial.
Lemma3.3. Assume Ω = С, then he L^ is harmonic iff h is Ш-measurable.
Proof. If h = Г*А, then T*h+^h and T*h+^0 imply Γ*Α+ ^ A+. Hence
£ = IIA ||«J -1 — h+ is superharmonic and therefore harmonic. Hence Г*А+ = A+.
If B= {A>0}, then
Γ*1β = Г*(Нт„(лА+ л 1)) = limnT*(nh+ л 1) ^ Нт(лЛ+ л 1) = 1B.
Now we can again infer that T* 1B = 1B and В е if. Applying this argument to A
— α we obtain {A > a} = {(A — a) > 0} e if for all a.
For the converse conclusion approximate h with ^-measurable step functions
and use that Be if implies Γ* 1β = 1β. α
In the general non conservative case we shall have to study the influence of the
dissipative part in C, the contribution of lD/to the limit in C. For this we shall
need (for Ε = С) the operator
(3.8) tf£ = /£ Σ (TIEC)\
v = 0

Brunei's lemma and the identification of the limit 127
As TE is a positive contraction in L1 for each η and Ее si, (3.3) implies
η
\\h Σ {TIe^Y/Wi й II/Hi f°r aU/e L\. Therefore, HE is a positive contraction
v = 0
in Lt. Using #£, the identity (3.5) can be written as
(3.9) 4>Ef=<HEfiE} = <HEfJ}
and we see that ψΕ = Щ H.
The operator #£ maps the mass /to the place where it first arrives in Ε under
iterated applications of T. Recall that gμ denotes the measure ν with dv/άμ = g.
ν | ^ will be the restriction of ν to the σ-algebra ^.
Theorem 3.4 (Neveu-Chacon). If Τ is a positive contraction in Ll9 fe Ll9 and
ge Uu then
ιζϋ-wam —*.>—·
If μ is a probability measure, the Radon-Nikodym derivative on the right side is
the ratio E{Hcf\c^)IE{Hcg\c^) of the conditional expectations. This is the most
common way of writing the limit, but it does not simplify matters because a
conditional expectation is a Radon-Nikodym derivative.
Proof, (a) We at first assume Ω = С Then Hc is the identity. If A = {S^g > 0},
then Sng tends to oo on A. For all he L\(A), hn = h л Sng tends to h and Thn
rg Sn+1g vanishes in Ac. Hence, Th — lim Thn belongs to ЕХ(А). This implies
AeW. For any pair a < b we now consider the set В = {a < lim Snf/Sng < b}.
On 5, Sn(f— ag) is positive infinitely often. Brunei's lemma implies
Put Я' = {т/>я = 1}, then B^B' e<€ and 1B ^ 1B, = T* 1B imply tpB ^ 1B„
Hence T/>B = \w and J (/— ag) \Β,άμ^ 0. The same argument with В replaced by
BnG and Ge4! shows J(/— ^)10ηΒ^άμ ^ 0. As C? was arbitrary, a
й ά{/μ | <0/</(g/i I *) on 5'·In the same way one can show ά(/μ | <0/</(g/i I *) ^ 6
on 2?'. As a<b were arbitrary this proves the theorem for Ω = C.
(b) Let us write L(/ g) := lim Snf/Sng. We now consider the case Ω Φ С We
may assume/^ 0. Choose some fixed/? e L\ with {p > 0} = С If/= r + я and
f = r + Ts, then £„/' = Snf+Tns- s. Using lemma 2.3 we therefore see that
f^f implies L{f\ p) = L(/ /?) on С By induction this remains true for/22*/'.
Set/W = lc7?/ Then/W £ T?fwaf^ 7?/imply L(/w,/>) £ L(//>) on С We
may already apply the identification of the limit in the conservative case to fm
because Г restricted to Lx (C) is conservative and/m e Lx (C). This has the
following consequence: if/m is an increasing sequence in L^C) tending to some
f„ e LX(C)9 then limmL(/m?jp) = L(f^p) in С Therefore fm = 1C7?/T Hcf
yields L(Hcfp)uL(fp) in С

128 Positive contractions in Ly
On the other hand,
ic Tkf= icTk{icf) + ic тк~1 (Tic)f= ... = ic T\icf) + ic тк-' (1с(Т1ссул
+ Ic Tk-2(Ic(TIcc)2f) + ...+ Ic(TIccff implies
"ς icTkf= "ς Σ icTv{ic{TiCcf-vf)й "ς icTvHcf.
fc = 0 fc = 0 v = 0 v = 0
Hence SnHcf^ Snf for all η on С Together with the inequality proved above we
obtain L{Hcf,p) = L{f,p) in С Similarly L{Hcg,p) = L{g,p) in С
On Cn{Soog>0} we may write S„/(co)/Swg(co) = (SJ(d)/Snp(a))).
(Sng(co)/Snp(co)) l for η large enough. Hence L(f,g) = L(f,p)/L(g,p).
Similarly, L(tfc/, Hcg) = L(Hcf,p)/L(Hcg9p) on Cn {S„ tfcg > 0}
= Cn{Saog> 0}. Combining our results we have proved L(/, g)
= L (Hcf, Hcg) and may complete the proof by applying the conservative case to
Hcf^HcgeL^C). π
Changing the reference measure. If μ' is a σ-finite measure with μ' ~ μ the results
above can also be expressed in terms of Lx{^). μ' ~ μ is equivalent to the
existence of a strictly positive measurable ρ with μ'{A) = \\Αράμ9 (A e si).
Clearly, Lx (μ) = Lx (μ'). Exploiting the isomorphisms Lx (μ) <-* Lx (μ) and
Li (μ') ++ L\ (μ') the operator Τ in Lx (μ') corresponding to Tin Lx (μ) has the
form T'f = /Γ1 T(f · p). (f e Lx (μ') of peLx (μ)). It is trivial to check that
Τ has the same conservative part and the same system of absorbing sets. Clearly
(T'Yf = p'1 Tn(f - p). This allows us to pass always to a probability measure
/ι'~/ι.
3. Convergence of Cesaro averages. The reader may have wondered why we have
studied the question of convergence of Snf/Sng before thinking about the
more obvious question of convergence of Anf= n~x Snf. There are several
reasons. If there exists an integrable strictly positive ρ with Tp = p, we have Anf
= pSnf/Snp and we see that the two questions are equivalent. We shall even be
able to derive the convergence a.e. of Anf if there exists only a strictly positive ρ
with Tp S P, which need not be integrable. In this case, however, the limit of Anf
frequently is 0 and the convergence of Snf/Sng may contain more information.
Finally: if no extra condition is added, the averages Anf need not converge a.e.,
(§3.6), and we obtain only stochastic convergence of Anf, (§3.4).
Passing to the reference measure μ' = ρ · μ the a.e.-convergence in the
following theorem may already be deduced from theorem 1.7.3 even under the weaker
assumption Tp ^ p. But the full assumption Tp — ρ allows to give simple
expressions for the limit.
Theorem 3.5 (Hopf). Let Τ be a positive contraction in Lx (μ) for which there
exists a strictly positive measurable pe Lx with Tp — p. Then

Brunei's lemma and the identification of the limit 129
CO for ft £i(/4 Anf converges a.e. to ρ · d(f· μ\<β)/ά(ρ · μ\<β)\
(ίί) for g with pg ε Lx (μ), in particular for g e L^,
lim A%g = d((p'g)^)/d(p№) a.e..
Proof (i) Apply theorem 3.4 to g = ρ and observe Ω = С
(ii) Let μ! — ρμ. Then pgeLx^) is equivalent to geL1(^). For geL^,
\{T*g)d^ = $p(T*g)dμ = \{Tp)gdμ = \pgdμ = \gdrf. Hence Γ* preserves
//-integrals and may be extended to a positive contraction in Lx (μ'). In
particular, S%g is well defined for all g e Lx (μ')· As Γ is conservative H = Γ* 1. Hence
A*g — S*gjS*^. As H is r*-invariant and integrable, Γ* is a conservative
contraction in Lx (μ'). Check that is has the same absorbing sets as Τ and apply
theorem 3.4. α
We now come to a rather general ratio ergodic theorem.
Definition 3.6. A sequence (pk), (k — 0,1, 2,...), of non negative measurable
functionspk on Ω is called admissible (for a positive contraction Tin Lx) if 7/?k
= Л+i holds for all к ^ 0.
The/?k need not be integrable. If/? is any measurable and non negative function
the sequence^ = Tkp is admissible. If ρ is a non negative function with Tp ^ p,
also the sequence рк= р is admissible.
Theorem 3.7 (Chacon [1963]). Lei (/?fc) be admissible for a positive contraction Tin
Lx andfe Ll9 then
Σ A
fc = 0
oo
converges a.e. on { Σ Pk > 0}·
fc = 0
Proof We may assume/^ 0. We may also assume Ω = С, since 5^/converges in
Z>, С is Γ-absorbing, and the sequence (lcPk) is again admissible. Let e be a
strictly positive integrable function. Set #k = ά(ρ]ίμ\<^)/ά(€μ\(^). (This Radon-
Nikodym derivative is well defined on the largest set (mod μ) where the restriction
of ρ1ίμ to ^ is σ-finite. We put qk = oo on the complement of this set). For
Ее <в, <pk, 1£> = <pt, Г* 1£> = <7>fc, 1£> й <ft+i, h>· It follows that the
sequence #k is increasing. If / ^ 0 is a fixed integer consider a sequence gve Lb with
{#v > 0} = {/>, > 0}, increasing to pt.
For any ν we have on {pt > 0}

130 Positive contractions in Lx
*-l ^,: $7"/
lim sup £>„(/, (л)) 5= lim sup—£ 5Ξ lim sup
Σ л
n->oo V1 ^ j-^οο *^/ί?ν
ά{/-μ\^) ά{/-μ\<£) ά{εμ\^)
d[gy-\i\<e) ά{βμ\<£) d(g^\
If ν tends to infinity ί/(£νμ|#)/ί/(βμ|#) tends to #г and we obtain
dwm ι
(3.10) lim supD„(f,(pJ)u
ά(βμ\?) q>
In particular, the limit of D„(f,(pk)) exists and is 0 on {qx = oo}, where
q^ = lim ^г. It remains to prove the convergence on the sets AK = {qx ^ K) for
K> 0. As these sets belong to # we may assume Ω = AK. Put
η oo
w0 = A» wi = Pi ~ TPi-n wn = Σ ",·, w = Σ "г-
i = 0 i = 0
The щ are non negative. It follows from J Tpt-X = J A-i that || wJh = \\ρη\\χ. As
^ is bounded we have \\pn\\x = 11^„11x й Ik^Hi й Ще\\х < oo, and w is inte-
grable. For all Ee4>,
ί ui = 5 Pt-5 TPi-i = ίPi ~ iPt-i ·
Ε Ε Ε Ε Ε
Hence ά{\νημ\4!)Ιά{βμ\4!) — qn, and ά(\^μ\(^)/ά(6μ\(^) = q^.
By induction рк = 7>к_! + (рк - Трк_х) yields
Л = Ткр0 + У ^v(A-v - 7>fe-i-v)? (* ^ 1).
v = 0
Summing from 0 to η we find
ΣΛ=Σ ГЧ + "Σ Τ\Ρί - Τρο) + "Σ 1*(ρ2 - Τρί)
к=0 к=0 к=0 к=0
+ ... + (р„-Тря-1)й Σ Tkw„u Σ Tkw.
k=0 k=0
Appealing to theorem 3.4 again we obtain
lim inf Dn{f, (A)) ^ hmSJ/Snw = (ά(/μ\^)/ά(βμ\^) · q~\ D
n->oo
It is clear from the proof of this theorem that
ά{βμ\<β) q^
where q^ = lim#k and qk = ά((1€ρ1ί)μ\^)/ά(βμ\(^).
к

Brunei's lemma and the identification of the limit 131
Corollary 3.8 (Dunford-Schwartz [1956]). If Τ is a positive Lx — L ^-contraction
Anf converges a.e.for all fe Lv
Proof. Take pk = 1. π
Definition 3.9. A non negative real valued measurable (not necessarily inte-
grable) function ρ is called subinvariant (or T-subinvariant) if it satisfies Tp ^ p.
If Tp = p, it is called invariant. Similarly, a measure ν = ρ · μ is called
subinvariant (invariant) if ρ is subinvariant (invariant).
If a subinvariant σ-finite measure ρ μ = ν ~ μ exists, then we may use μ' = ν as a
new reference measure and T'g = p~xT(gp) is an Lx — L^-contraction in
Lx (μ'), because T'i =p~xTp<L 1. An application of Corollary 3.8 to Τ shows
that ^„/converges μ-a.e. for all/e Ι^χ(μ).
It follows from $T*hdtf = \{Τ*Η)ράμ = \Η{Τρ)άμ й \1ιράμ = \hdyl,
(h ^ 0), that Γ* is an Lx — L^-contraction in Lx(^). Consequently also A*h
converges a.e. for all he Lx {μ'). Τ* in Lx (μ') is called the dual Markov operator
to Τ
Lemma 3.10. If a measurable real valued ρ ^ 0 is Tsubinvariant, then ρ — Tp
holds on C. In this case ρc = \cp is invariant. If a Tsubinvariant, strictly positive
real valued ρ exists D is T-absorbing.
Proof. Choose some he L^ with </?, A> < oo and with {h > 0} = C. For all л,
<P ~ Tp, "Σ T*lhy = </>, h} - </>, Г*"А> й <p9 h} < oo.
i = 0
As 5* A diverges on С we obtain ρ = Tp on C. As С is Γ-absorbing and
Tpc ύ Ρ the function pc is subinvariant and the second assertion follows from the
first. But then T{\Dp) must vanish on C, and thus, for all feL\(D),
Tf= Нтп(Г(/л n\Dp)) must vanish on C. α
Lemma 3.10 shows that the assumption of existence of a ^-invariant ν ~ μ is
fairly strong because it excludes any "influence" of the dissipative part in С We
therefore mention that the a.e.-convergence of ^4„/for all/e Lx follows also from
the weaker assumption of existence of a subinvariant ρ with {p > 0} = С To
prove this we may assume/^ 0. Now the convergence of ^4„/to 0 is trivial on
D и {£,/ = 0}. On С η {£,/> 0} theorem 3.4 implies lim,^ Sn(Hcf)/Snf= 1.
We can therefore replace / by Hcf and apply the result proved above to the
restriction of Τ to Lx (C) and to Hcf
The limit is determined by (3.11) with pk = p. But there is a more explicit
description: Let us say that a finite invariant measure exists on a set A if there is
some invariant q e L\ with {q > 0} => A. By an exhaustion argument we can find

132 Positive contractions in Lx
a sequence of sets Al9A29... and invariant functions qx,q2, ...eUx with
At = {?i > 0} such that no finite invariant measure exists on any В of positive
measure in the complement of the union С of Al9A29 The function
00
ρ = Σ 2"1 ll^-llj"1^ is Γ-invariant and integrable and has support {p > 0} = C.
i = l
С is the maximal set on which there exists a finite invariant measure. Clearly
С с С. С will be studied more carefully in the next section.
Lemma 3.11. If a поп negative measurable ρ is T-subinvariant, its support {p > 0}
is T-absorbing.
Proof. Forfe L\ with {/> 0} с {ρ > 0} the sequence/л пр increases to/and
Tf=\imnT(fAnp). α
Theorem 3.12. Let Τ be a positive contraction in Lv Assume that there exists a
real valued subinvariant ρ ^ 0 with {p > 0} = C. Then Anf converges a.e.for all
integrable f. The limit vanishes in the complement of C. Set f = X^H^f — Hcf
Then the limit equals ρ · ^(/μ|^)/^(/?μ|^) on С.
Proof As С is contained in С and Γ-absorbing, also C\ С is Γ-absorbing. Using
this fact it is not hard to verify the heuristically obvious identity X^n^f— Hcf
The formula for the limit on С therefore follows from theorem 3.5 and it remains
to show that the limit vanishes on C\ C. We may assume Ω = C\ C. Set pk = ρ
again. The restriction οϊρ4μ to ^ has no sets of positive finite measure, because
for any such set^l the function pklA would be integrable and Γ-invariant. It
follows that the qk in the proof to theorem 3.7 are = oo and hence q^ = go. By
(3.11) linw Ли/= linw £„(/, (pk))-p = 0. G
Remarks on ^-convergence. In the dissipative case the sequence Anf for fe Lx
may or may not converge in Lrnorm; see § 4.3.
In general, if ^„/converges in norm to some/then/is Γ-in variant. If Γ is
conservative Γ*1 = 1 implies \Anf= J/. Hence, for/e L\ (C) the set {/> 0} is
contained in C. As C\ С is Γ-absorbing, norm convergence of Anf cannot hold
for any fe L\ (C\ C) with / Φ 0.
On the other hand Lx-convergence of Anf for/e LX(C) is a rather simple
result; see §2.1. It follows also from a.e.-convergence together with uniform
integrability.
Definition 3.13. A sequence (/„) in Lx is called uniformly integrable if, for each
ε > 0, there exists a gee L\ with \{\fn\— g^* άμ<ε for all n.
To prove the uniform integrability of {Anf)n^x for fe LX(C) one can take
ge = KP with к large enough since (An\f\ - kp)+ = (An(\f\ - kp)) +
uAn{\f\-kp)\

Brunei's lemma and the identification of the limit 133
We also prove a more general lemma valid even for non integrable p:
Lemma 3,14. Let ρ ^ 0 be T-subinvariant for the positive contraction Τ in
Ll9 h e L^({p > 0}) αηά\Ηράμ < oo. Then, for any fe Ll9 the sequences {Tnf)h
and (Anf)h are uniformly integrable.
Proof We may assume/^ 0 and h ^ 1. Set Ε = {p> 0}. Forfe Lx (E) we may
argue as above: The uniform integrability of {h Tnf) follows from {h Tnf— khp)+
uh(Tnf-kp)+uh(Tnf-kTnp)+uhTn(f-kp)+. For general / put/v
00 П
= IE(TIEyf. Then Σ /, = HEfe Lx and lETf= Σ T»~VV.
v=0 v=0
As the/v belong to Lx (E) each sequence (Tnfv)n^0 is uniformly integrable, and
hence also each sequence (?n,v)n^0 with g„tV = 0 for ν > η and gnv = Tn~vfv for
ν <; n. Now, for all AT ^ 1,
v = 0 \ = Κ + 1
As the integral of the second summand can be made arbitrarily small uniformly
in η by a large choice of AT and the first summand is uniformly integrable as a finite
sum of uniformly integrable sequences the sequence A 7™/must be uniformly
integrable. But then this must be true also for (hAnf). α
If/is an initial distribution of a Markov process for which the transition
probabilities determine Γ, then (Tkf 1A} is the probability of a visit in A at time k. The
lemma can be combined with the Dunford-Schwartz ergodic theorem above to
deduce the convergence of Cesar о averages of < Tkf 1A}, when A has finite subin-
variant measure.
For functions / with \/άμ = 0 there is Lx-convergence in the conservative
ergodic case:
Theorem 3.15. If Τ is a positive contraction in Lx andfe Lx (C) then \\Anf\\x -> 0
holds iff J 1Ε/άμ = Q for all EeV.
Proof We may assume Ω = С The only if part is obvious because (Tkf 1E}
= if T*fcl£> = </, 1E> holds for all Ее <€. Now assume </, 1E> = 0 for all
Ee4>. Then </, h} = 0 for all ^-measurable he L^. By lemma 3.3 he L^ is <€-
measurable iff it is harmonic. Using T* h = h о (g, T* h} = <g, h} Vge Lx о
(Tg — g,h} = 0 VgeLi we see that h is harmonic iff h is orthogonal to
(T— I)LX. Hence, by Hahn-Banach,/belongs to the closure of (T— I)LU and
for such/the convergence M„/Hi -* 0 is simple to prove, α

134 Positive contractions in Lx
Notes
Lemma 3.1 is due to Akcoglu-Chacon [1970a]. The present simple as yet
unpublished proof has been given by Giroux. In the original formulation of Brunei's lemma
the set {ω: S„f(co) > 0 for infinitely many n) was replaced by the smaller set
{ω: sup„ TkSnf((D) > 0 for all к ^ 0}. The sharper formulation was obtained by Akcoglu
[1965]. Brunei [1966] has pointed out that it can also be derived from his formulation. In
spite of alternative proofs by Meyer [1965], Garsia [1967] and Revuz [1975], Brunei's
original proof may still be the shortest. A continuous time version of Brunei's lemma has
been given by Akcoglu-Cunsolo [1970]. Neveu [1961] proved the identification of the
limit in the case Ω — С and stated the general case. Independently, Chacon [1962a]
proved the full result. The proof with Brunei's lemma is due to Meyer [1965a].
Akcoglu-Sharpe [1968] and Akcoglu-Chacon [1970] have developed an identification
of the limit also on D. They study the partial order in L\ defined by/-< g iff there exists an
и^О with f^+g. A g ja/ is called asymptotically invariant if, for all fe L\, Xf(A)
= lim(gju) (A) exists when g ranges through Pf-= {ge L\:f^g}. It is shown that the
family Σ of asymptotically invariant sets is an algebra and that Xf is additive. For g e L\
the set G -·= {S^g > 0} belongs to Σ. h = lim„ Snf/Sng exists a.e. on G. The sets {h ^ a)
belong to Σ except for countably many a, and, for A e Σ with A a G, one has \A h dXg
= λ/(Α). On С the algebra Σ agrees with ^ and λ9(Α) = (Η^)μ(Α). Σ may be fairly rich
even in the case Ω = D.
Berk [1968] has studied the identification of the limit on C, and Akcoglu-Cunsolo
[1970a] on all of Ω for the continuous parameter situation. For a general continuous
parameter ratio theorem see Terrell [1972a]. Tsurumi [1954], [1958a] proved Hopf type
and ratio theorems under weakened stationarity assumptions.
Lemma 3.10 appears in the systematic study of subinvariant measures by Feldman
[1962]. Theorem 3.15, observed independently by Sucheston [1967a] and Krengel
[1966], is essentially equivalent to a factorization announced by Chacon [1962b]: If Γ is
conservative and pe Lx strictly positive, any integrable / splits into a function hp with
T(hp) = h(Tp) and an element of cl(T— V^L^ see also theorem 7.5.
Theorem 3.12 is a special case of results in Deriennic-Lin [1973]. Akcoglu-Chacon
[1965] showed that conservative positive contractions in Lx which are contractions in
some Lp, (1 <p< oo), are contractions in all Lp, (1 <Ξ ρ <Ξ οο); Μ.Μ. Rao [1966] has
related interpolation theorems in Orlicz spaces.
It is easy to see that the dual Markov operator constructed for a positive contraction Τ
admitting a strictly positive subinvariant function p, has the same conservative part as Τ
and the same σ-algebra ^.
Induced operators. For a positive contraction Τ in L1
t^e = (iet) Σ ihcTf
k = 0
is called the operator induced by Ton Ε e s/. If Tis the contraction in Lx corresponding to
a null preserving τ, ΤΙ*£ corresponds to the induced transformation τΕ. It is simple to check
that T_+E is a positive contraction in L1 (E) with conservative part EnC, and that the 7^£-
absorbing subsets of Ε are just the restrictions to Ε of the Γ-absorbing sets. One has
He = h + T_+EIEc.
For a more thorough discussion we refer to Horowitz [1968a] and Brunei [1971], and
for a systematic study of related operators to Neveu [1972]. Simons and Overdijk [1979]
investigate induced processes and the existence of "sweep-out-sets".
Scheller [1965] proved a generalization of the Kac recurrence theorem: For

Existence of finite invariant measures 135
сое E, T*(IEcT*yiE(co) is the probability that the first visit to E, given a start in ω,
00
happens at time (v + 1). Thus, νΕ{ω) = Σ 0 + l)/£ ^* (/£c Τ*)ν1Ε(ω) is the expected first
v = 0
return time. Scheller showed (both directly and via the Markov shift in Ω+): If μ = Τ μ is a
finite invariant measure, J£ γΕ(ω)άμ = μ(Ε*), where Ε* = {ψΕ = 1} is the smallest T-
absorbing set containing E.
Supplement to § 1.6. An application of the Brunei function enables us to complete the
proof of proposition 1.6.8 also for non invertible τ:
Proof of Proposition 1.6.8. Let Τ be the contraction in L1 given by Tf = f°x. As τ
is conservative Τ must be conservative. # is just the family of τ-invariant sets. As
00
A* = (J T-iv4is τ-invariant we may assume Ω = A*. As Τ is conservative Τ*ψΑ = трл
i = 0
and 1рл is ^-measurable. Hence ψΛ = 1. The sets {^ = /ι + 1} = ^ητ 1Acn...
ητ~ηΑ€ητ'(η+1)Α with и ^ к form a disjoint decomposition of Ak-= Ar\x~iAcr\...
пт~кАс when к ^ 1.
The proof can therefore be completed by the following computation:
J s = <i,s> = <v*s> = < Σ (/^*)* W> = <1л, ς (TiAC)kg>
A* k = 0 fc = 0
Л к = 1 A A k = l Ak
= is+ Σ Σ ί g°tk = $g+ Σ Σ ί Γ^
Л к=1 п = к {rA=n+l} A n = l k = l {rA=n + l}
oo m —1 r^-1
= U+ Σ ί Σ£°τ* = ί Σ ?нк- α
Л m = 2 {гл = т} fc=l Л fc = 0
§ 3.4 Existence of finite invariant measures
In this section we discuss necessary and sufficient conditions for the existence of
finite invariant measures for positive contractions in Lx and for nonsingular
transformations of a measure space. An application is the proof of the stochastic
ergodic theorem.
1. Banach limits. Recall from §1.1 that a Banach limit L is a positive linear
functional L on 4» invariant under the shift Θ: (x0, xi9...) = χ -* θχ
= (*i? *2> ···)> and satisfying L(l) = 1, where 1 = (1,1,1,...).
The properties of Banach limits readily imply ||L|| = 1 and
(4.1) lim inf xn ^ L(x) rg lim sup xn.
Theorem 4.1. (a) Banach limits exist.
(b) For fixed χ the maximal value of Banach limits sup {L(x)\ L is a Banach

136 Positive contractions in Lx
limit) is given by
n-l
(4.2) M(x):= lim (supи-1 Σ xi+j)·
n->oo j i = 0
(c) For fixed χ the values L(x) of all Banach limits agree and are equal to s iff
n-l
lim n~x X xi+j = s uniformly in j,
n->oo i = 0
n-l
Proof Set cn = sup и-1 Σ xt+j- F°r each К m one has ckm rg cm. Thus
j i = 0
(r + km)cr+km ^ re, + kmckm ^ rcr + kmcm.
Dividing by r + km and letting A: tend to oo we obtain lim sup^^c,..^ ^ cm.
Since this holds for r = 0,1,..., m — 1, lim sup c„ ^ cm for each m, and hence
lim sup cn ^ lim inf cm. Therefore the limit in (4.2) exists.
Let 4° be the subspace of 4 consisting of the convergent sequences. L0(x)
= lim„ xn defines a linear functional on £®. Μ is a sublinear functional on 4,, i.e.,
Μ (χ + у)й Μ (χ) + Μ (у) and if α ^ 0 then Μ (αχ) = αΜ(χ). As L0 (χ) = Μ (χ)
holds for χ g /<£, the Hahn-Banach theorem implies the existence of a linear
functional L on (^ which agrees with L0 on /° and for which L(x) rg Μ (χ) holds
for all xe^. Obviously, L(l) = L0(l) = 1. For χ = (x„) with x„ ^ 0 for all и,
L(x) = - L(-x) ^ - M(-x) ^ 0. Finally
\M{x — θχ)\ = \limnsupjn~1(xj — xj+n)\ rg limnH_1 2supjxfc| = 0
together with the symmetric equation M(0x —x) = 0 imply L(x) — L(Qx)
= L(x — 0x) rg M(x — θχ) = 0 and L(0x) — L(x) rg 0. Thus, L is a Banach
limit.
To prove (b) we make use of the fact that, for fixed χ e 4>\^? the extension
procedure in the Hahn-Banach theorem may be started by assigning to L(x) any
value in the interval
[sup{(-M(-x-^)-L0(;^e^
To show that, for any χ e 4>, there is a Banach limit L* with L* (χ) = Μ (χ), we
may assume χ e 4>\£?· If £* is an extension of L0 with L*(x) = inf {(M(x + >>)
-L0{y)):yet*}9 then L*(x) = M(x) since M(x + y) = M(x) + L0(jO for
m-l
>>e^. Conversely, letL be any Banach limit. Set znm) = m~x Σ xi+n>
i = 0
(n = 0,1, 2,...). Then L(x) = L(z(w)) ^ supnznw). With limw supn zww) = M(x)
this yields L(x) rg Μ (χ) and completes the proof of (b).
n-l
(c) follows from (b) since now — M( —x) = lim (inf n~l Σ xi+j) is the
minimal value of Banach limits, α η j t=o

Existence of finite invariant measures 137
2. Existence of finite invariant measures. Now let Γ be a positive contraction in
Lx = Lx (Ω, sf9 μ), where μ is a σ-finite measure on (Ω, si). Recall that Γ induces
a transformation ν -> 7V in the space Lx of finite signed measures ν <ξ μ: If ν
= /· μ (i.e.,/ = dv/άμ), then 7V = (7/) · μ. The question of existence of a finite
invariant ν e Д is equivalent to the question of existence of a Γ-invariant/e Ux.
If μ' is a probability measure with μ' ~ μ, then Lx (μ') = Lx (μ). Passing to a new
reference measure we may and shall therefore always assume μ(Ω) = 1 in this
section. Recall the notation <v? A> = J hdv, Lp(B) = {he Lp: {H0}c B). The
starting point of the investigation is
Theorem 4.2. Let Be si. There exists anfe L\ with В а {/> 0} and Tf = fiff
(4.3) inf <μ, Γ*ηA> > 0 /or a// A wifA 0 Φ h e L+(B).
Proof. The easy half is to show that (4.3) is necessary: We may assume μ (Β) > 0.
If there exists an/e L\ with {/> 0} з jB and Г/= /, then v=f· μ satisfies Γ" ν
= v for all л. For any heL^(B) with Α Φ 0 there exists an g>0 with
(v,b1)> 2811/11!. Now <v? T*n(h л 1» = <ГЧ йл1)> 2ε||/||1 for all n.
Set 4 = {Г*Й(Ы)^£}, then Г*И(Ы)^1 implies v(^n)' 1 + ev(A$
^ <ν?Γ*η(ΑΛΐ1)>>2ε||/||1? and hence v{An)^&\\f\\x for all и. By ν <| μ
there is a <5 > 0 with μ(Αη) ^ <5 for all и. Consequently, ίηί„<μ, Γ*ηΑ>
^ ίηί„<μ, Γ*η(Α л 1)> ^ ε<5 > 0. Now assume (4.3). Take any Banach limit L
and define a linear functional ρ on L^ by ρ (Α) := Ζ,(«Γημ, A»). Observe that
ρ (A) ^ 0 for A g L+, and that ρ(Γ* Α) = ρ (A). Define, for A e Lj,
00 00
v(A)==inf{E (?(*.):* = Σ *„*„€£+}.
n=l n=l
For A, A', A", and (Aw) e Lj with A = A' + A" and A = Σ^„ there exist sequences
A;, /£ g L+ with A' = ΣΑ; and A" = Σ A»: to see this start with A'x = А' л А1? A'/
= hx — h\ and continue with A — A1? A' — h\ and A" — A'/, etc. Hence v(A) ^ v(A')
+ v(A")·
To show that ν is "σ-additive" on L+it now suffices to verify that A = Σ gm>
m = \
oo
(gm, A g L+), implies ν (Α) rg Σ ν (gm). By the definition of ν there are sequences
m = l
hn,meLt with gm = £ hn<m and Ze(*..J < vfej + 2-<"+4>ε. Now v(A)
n = l η
00
^ Σ Q(hn,m) й Σ v(£m) + ε because A = Σ Aw>m. As ε > 0 was arbitrary ν is
n,m m=l n,m
σ-additive and defines a finite measure ν <ξ μ by v(^l) = v(l^). Set/= dv/άμ.
We now want to check В с {/> 0}. Otherwise there is some А а В with v(^l)
oo
= 0 and μ (A) > 0, and then we can find sequences (Amn) e L+ with 1A = Σ ^m>„
n = l

138 Positive contractions in L1
00 00 00
and Σ Q(hmfrl) < m'1. If k(m) is so large that Σ Σ <μ> ^m>n> < <μ, A>,
n = l m = l n = fc(m) + l
fc(m) fc(m) fc(m)
then 0 ^ A* ==inf { Σ hmJ * 0. For all m, ρ(!ι*) ^ρ(Σ hmJ = Σ Q(hm,„)
m и=1 и=1 и=1
< m"1, and hence ρ (A*) = 0. The function A* belongs to L^(B) since A* rg 1л
and Aa B. Now ρ (A*) = 0 contradicts (4.3).
The 0-invariance of L implies ρ(Γ*Α) = ρ (A). For each sequence (A„) with A
= Σ hn we have ν(Γ* Α) ^ Σ Q(T*hn) = Σρ(Λ„); we therefore obtain ν(Γ* A)
и=1 и=1
rg ν (Α); ν is sub-invariant. Next, observe that В is contained in C: If В has a non
00
empty intersection with the dissipative part £> = { Σ Τηί <co} there exists an
n = 0
00 00
AeL+with{A>0} = 5nD Φ 0forwhich J( Σ Τηί)·Ηάμ= Σ <μ,Γ*ηΑ> is
finite. This is impossible by (4.3). n=0 n = 0
Put ν (A) := v(^4 η С). The density/= Jv/φ = lc/is strictly positive on B. As
Tv has support in С by the Γ-invariance of C, and Γν ^ Γν rg v, we obtain Γν
rg v. But then ν is invariant since the support of ν is contained in С ν is the desired
invariant measure, α
We now define weakly wandering functions which will help us to deduce stronger
and more useful conditions.
Definition 4.3. A weakly wandering function A is an element of Lj for which there
exists a strictly increasing sequence 0 = k0 < k1 < ... of integers with
00
(4.4) || Σ r*kvA||00<oo.
v = 0
If even the stronger condition
(4.5) sup || Σ 7τ*^-·Ά||00<οο
holds, A is called m-weakly wandering.
Lemma 4.4. IfgeL^ has the property
(4.6) Hminf<r4,g> = 0
и->оо
then for any ε > 0 there exists an m-weakly wandering function A rg g with
(0 \^-Η)άμ<ε and (ii) {A > 0} = {g > 0}.
Proof We may assume \\g\\1 ^ WgW^ <; 1. (4.6) is equivalent to
(4.7) lim inf <Tnf g} = 0 for all fe L\.

Existence of finite invariant measures 139
For a number b with 0<b< 1, which will be specified later, there exist
sx > s2 > s3 > ... > 0 with
00
1 + 6= π(ΐ + βν).
v = l
Using (4.6) we find/i >0 with ||(/+ T^g^ = <1,g> + (Thi,g} й 1 +βι·
Applying (4.7) with/= (/+ Th)i we find i2 > 0 with ||(/+ T*h) (/+ T^g]^
^ (1 + ej) · (1 + ε2). Continuing in this way we inductively define a sequence /l5
i2, i3, ... of positive integers with
||(/+ Г*'0(/+ Γ*ί2)... (/+ Γ*^^ ^ Π (Ι +εν).
v = l
The sequence gin) = (/ + Γ**1) (/+ Γ*1'2)...(/ + r*lV)g increases to a limit g(oo)
with \\g{00)||i <; 1 + 6. Our first aim is to show that the function cp = (2g- g{co)) +
η
is m-weakly wandering for the sequence kn — X /v, (л ^ 0). From
v=l
φ ^ (2g - (/+ T*in)g)+ = (g- T*ing)+ it follows that (pug- T*ing holds on
{φ > 0}, and hence that φ + Τ*ίηφ й φ + Γ*1^ ^?ν Γ*^ ^ HgH^ ^ U. The
inequality
(4.8) φ(Β»*>:=φ+ Γ*^φ + :τ*(ί* + ί* + ι)φ + ... + Γ*(ίί' + ίί' + 1 + ··· + ίη)φ ^ 11
is therefore valid for 1 ^ ρ = л. Once it is proved for 2 < ρ ^ n, we can derive
φ<".ρ-ι>^1: Note that
φ ^ (2g- (/+ Γ*'*"1) (/+ Γ*'*)... (/+ T*in)g) + .
This implies that, on {φ > 0}, the inequalities cp^2g-(I + Τ*ίρ~ι) φ{"'ρ) ^ 2g
_ ^ _ j*«p-1 ^(«.p) = g _ y*ip-1 ^(«.p) hold true. Consequently, φ*"»*"1) = φ
_l_ γ*ΐρ-ΐφ{η,ρ) <^ ^ Using backward induction we have verified (4.8) for 1 ^ /?
If 7 ^ 0 is fixed let / be the first index with kt *>j. Then, by (4.8),
00
у 2^*(λν-7)φ _ 7T*<kt~J)((Y) -|- V JT*(it+i+it + 2+...+iv)(y) < I) ^
fcv^J v=i+l
Thus, φ is m-weakly wandering. It is clear that 0 rg φ rg g. For b < ε the
inequalities
show that φ has all the properties requested for h except (ii).
To find h we have to modify the above construction. Let 1 > bx ^ b2 ^ b3
^ ... > 0 be reals to be chosen later. There exist ε1? ε2,... > 0 and iu i2, i3,...
> 1 with

140 Positive contractions in Lx
(4.9) Π(1+0^(1 + 6„), (и£1),
v = n
and
n + m
||(/+r*'-)(/+7'*'-+,)...(/+7'*l"+m)g|li^ Π (1+β,)» (w^l,w^0).
v = n
n + m
As above gi00» = lim Π (/+ T*^)g satisfies Hg^l^ й (1 + 6„) and g!,00» ^ g.
m->oo v = n
The function φπ = (2g — g^)+ ^ g again has the analogous properties as above:
\\g-(pn\\1ubnimd
00
v = n
If f is the first index with kt ^ 7, and t ^ л - 1, then Σ T*(kv~j) φη <; H as in the
above argument for φ. In the case t ^n — 2,
n-l
£ Τ^ν-])φη== £ ;τ*(*ν-./)φη+ £ Γ^^φ,,^Λ + Ι.
kv^J ν = ί ν ^ η
Now
А = (1-0Ф1 + Дв2--(и + 1)->||
is га-weakly wandering because
Σ T*kv~jh й 1
holds for all у'^ 0. A rg g follows from φη <; g, (n^l). For ^ < ε/2 we have
||g —A|li й \\g — Ψι Hi + 2_1ε<ε. It remains to show that {h > 0} = {g>0}
holds if the 6„'s are sufficiently small:
Because of μ(# — φη ^ и-1) ^ л|| g — φ„|| rg и&„ we can obtain
00
Σ nig — ψη = л-1) < 00 by choosing the 6„'s so that 6W < n~3. By the Borel-
n = l
Cantelli lemma then almost every ω belongs to only finitely many sets
{g-cpn^n'1}.
Consequently, a.e. coe{g>0} belongs to infinitely many of the sets
{ψη > 0} 3 {g > n~1}\{g-(pn^ n~1}. Hence {h > 0} = {g > 0}. D
Lemma4.5. If 0 = k0<kx< ... <kn are integers, and, for some he Lj and
η
ceR+, the inequality Σ T*kih rg с holds a.e., /Леи
i = 0
limsupMJAIL^/i + l)-^.
m->oo
In particular, if h is weakly wandering, lim M^IL = 0.

Existence of finite invariant measures 141
Proof. Simply observe that
m-l ko + m-1 fci+m-1
(« + 1)Σ T*lhu Σ T*lh+ Σ T*lh + ...
i = 0 i = ko i = ki
kn + m-l
+ Σ T^h + in + VbML·
i = kn
m-l η
= Σ γ*'(Σ T*k"h) + (n + i)k„\\h\\x
i=0 v = 0
£mc + (n + l)kn\\hL·. π
We are now in a position to prove a theorem which unifies many of the known
necessary and sufficient criteria for existence of a finite invariant measure.
Theorem 4.6. If Τ is a positive contraction in Lx (Ω, «я/, μ), there exists a
decomposition of Ω into two disjoint sets С, Д uniquely determined up to nullsets by the
properties
(i) there exists a p0e L\ with Tp0 = p0 and {p0 > 0} = C, and
(ii) there exists an m-weakly wandering h0 e Lj with {h0 > 0} = D.
The set С belongs to ^.
We still assume μ (Ω) = 1, but as both statements (i) and (ii) are invariant under a
passage to an equivalent measure μ' ~ μ (with a different p'0)9 the theorem is
equally valid for σ-finite measures μ. In the special case of a Markov chain with
countable state space, С is the set of positively recurrent states and D the set of
null states. С is called the strongly conservative part of Ω for T, or the positive part.
D is called the null part.
Proof. Set Ji = {geL^: lim MJgH^ = 0}. Let gn, (η Ξ> 1), be a sequence with
gne^ and μ^η >0)^η-.= sup{/ife > 0): ge Л). Then g0 = Σ (gn л 1)2""
n = l
belongs to Ji and we have
lim inf <Г"11, g0} = lim inf J Г*т£0Ф = О.
m->oo m-*oo
By lemma 4.4 there exists an m-weakly wandering h0 rg g0 with {h0 > 0}
= {g0 > 0} and we obtain (ii) by setting D = {h0 > 0}. If there exists a non zero
he Zj with support in €··=Ω\6 and liminf,,^ <ГИИ, h} = 0, there exists by
lemma 4.4 a weakly wandering А'ФО with {h > 0} с С. By lemma 4.5, A' belongs
to Ji\ and in this case g0 + A' would be an element of Ji with μ(#0 + A' > 0) = τ/
+ μ(Μ>ϋ)>η contrary to the definition of η. It follows that
liminfn<rwH,A>>0 for 0фАе£+(С), and therefore also that
infn <μ, T*nh} > 0 for 0 φ h e L+(C). By theorem 4.2 there exists a/?0 e L"i with
Tp0=Po and Сс={/?о>0}.

142 Positive contractions in Lx
If ρe L\ is Γ-invariant and he L+ is weakly wandering for the sequence
k0 < kx < k2 < ..., the finiteness of <p, Σ T*kih} = < Σ Τ*ρ, h} = (oo-p,h}
±=0 i=0
implies {p > 0} η {h > 0} = 0. Hence С = {p0 > 0}, and the same argument
implies the uniqueness mod nullsets of the decomposition Q = CuD. By
00
{ Σ Tlp0 = oo} = {oo -p0 = oo} = {p0 > 0} the set С is a subset of С. Се %?
i = 0
follows from lemma 3.10. α
Let us now show how theorem 4.6 yields some known criteria:
Corollary 4.7. Each of the following conditions is equivalent to the condition of
existence of a finite T-invariant ν ~ μ:
(F) There is no A with μ(Α) > 0 and lim MJ l^lL = 0;
n->oo
j + n-1
(DSN) For all A withμ{A)>Q, lim sup л"1 Σ <μ, Τ**ίΛ) > 0;
n-*co j i=j
η-ί
(IJ For all A with μ(Α) > 0; lim sup n~l Σ <μ, Τ*11Α} > 0;
i = 0
(I2) For all A with μ{Α) > 0, inf {μ, Τ*η1Α} > 0;
(B) cl((I— 7T*)L00)nLj= {0}, where cl is the norm closure.
Proof The existence of ν implies (I2) by theorem 4.2, and the implications (I2) =>
(Ix) => (DSN) => (F) are obvious. Now assume (F). If no finite invariant ν ~ μ
exists, D is non empty and there is a weakly wandering h0 φ 0. For small enough
β>0,Α'-= {h0 Ξ> β} has positive measure and 1A rg /?-1 h0 is weakly wandering.
By lemma 4.5 this contradicts (F).
It remains to prove the equivalence of (B): If there exists a strictly positive
fe L\ with Γ/= /, then </, (/- r*)g> = 0 for all g e L^ and hence </, A> = 0
for heH0:= cl((I- T*)LJ, while </, A> > 0 for 0 Φ h e L+. Consequently, in
this case (2?) holds. If such an/does not exist, consider A = {h0 ^ /?} as above,
00
and #! = {g g L*: r**g = g}. From Σ T*ki 1A e L^ it follows that <1 A, g> = 0
i = 0
for all /e #x = Щ. By the Hahn-Banach theorem {ge L^: <g,/> = 0 for
/e #J = #0. Hence 1леЯ0п1^ and in this case (2?) does not hold, α
3. The stochastic ergodic theorem. Now we show that theorem 4.6 can be used to
derive stochastic convergence for Cesaro averages if Γ is a contraction in Lx
which need not contract the L^-norm. A.e.-convergence and Lx-convergence fail
to hold in this generality.
Definition 4.8. A sequence {fk} of measurable functions on a σ-finite measure

Existence of finite invariant measures 143
space (Ω, «я/, μ) is said to converge stochastically to /^ if
μ(/1 η {\fk —f^ | > ε}) converges to 0 for all ε > 0 and all A with μ(Α) < oo.
Theorem 4.9 (Stochastic ergodic theorem; Krengel [1966]). If Τ is a positive
contraction in Lx of a σ-finite measure space (Ω, sf9 μ) then, for any fe Lx, the
averages Anf converge stochastically. The limit is T-invariant and vanishes in D.
For fe L\ it agrees a.e. with the pointwise liminf^4n/.
(All assertions but the last in this theorem remain true for non positive
contractions Γ in Lx if D is the complement of the strongly conservative part of Ω for the
linear modulus \T\\ see §4.1).
Proof By theorem 4.6 there exist a weakly wandering h0 e L+ and a Γ-in variant
p0 e Ux with {h0 > 0} = D and {p0 > 0} = C. By the Chacon-Ornstein theorem
the ratios Snf/Snp0 = Pq1 ^„/converge a.e. in C, and hence ^„/converges a.e. in
C. Because of Hcf= HcHcf theorem 3.4 implies that the limit on С is the same
as that of An(Hcf). As С is Γ-absorbing it is also the same as that of An(\cHcf).
In view of the Lx -convergence in Lx (C) the limit must be ^-invariant. Let us now
show stochastic convergence to 0 D. We may assume/ ^ 0. Let A be a set of finite
measure and Α (β) = An {h0 ^ β}. For given η > 0 there exists β > 0 with
μ(Α\Α(β)) <η/2. As ίΑ(β) ^ β~1h0 is weakly wandering, lemma 4.5 implies
Μί W) 11 * -> 0 and hence ит<^в/,1^)> = O.Fori* ^ Νη and Νη large enough
we have (Anf ίΑ(β)} < rl8/^ and then
μ{Αη{ΑΛ/>ε})£μ{Α\Α(β)) + μ(Α(β)η{ΑΛ/>ε})
£ηβ + ε-1<Αη/,ίΑ(β)><4
completes the proof of stochastic convergence to 0 on D.
As there is even a.e.-convergence on С the last assertion follows, too. α
4. Null preserving transformations. Many of the results on existence of finite
invariant measures have first been studied for contractions Τ given by a null
preserving!: Ω -* Ω in a probability space (Ω, s/, μ). Then Tv(A) = ν (τ-1 A) ana
T* h = h ο τ. In this case the concept "weakly wandering" has emerged in a
stronger and more intuitive form. A set We si is called weakly wandering if there
exists a sequence 0 = k0 < kx < k2 < ... of integers such that the sets x~ki W ъхе
disjoint. Theorem 4.2 implies that a finite invariant ν ~ μ exists iff there exists no
A with μ (A) > 0 and
(4.10) inf {μ{τ-*Α)\ 0 й к < oo} = 0.
If we start with such a set, repeated applications of the following lemma (with
sufficiently small ε/s) produce in the limit a weakly wandering set of positive
measure:

144 Positive contractions 1П L·^
Lemma 4.10. If A e si is such that for a finite sequence 0 = k0 < kx < ... < kr, the
sets T~kiA,(i = 0,..., r), are disjoint, and (4.10) holds, then for each sr>0 there
exists a set A' a A with μ{Α\Α) < гг and an integer kr+1 > kr such that the sets
T~kiA\ (i = 0,..., r +1), are disjoint.
Proof Let <5,.>0 be such that μ(Β)<δγ implies μ{τ~ιΒ) < (r + 1)_1ε,. for
ι = kr — kp (j = 0,..., r). Choose kr+1 > kr such that μ(τ~(*Γ+ J~kr)A) < Sr. Take
A' = A\(j T-(kr+1~ki)A. π
i = 0
We have proved:
Theorem 4.11 (Hajian-Kakutani). A null preserving τ in (Ω, s/9 μ) admits a finite
τ-invariant measure ν ~ μ iff there exists no weakly wandering set of positive
measure.
Notes
The existence of Banach limits is due to Banach [1932], Theorem 4.1 (b) to Sucheston
[1964], (c) to Lorentz [1948]. The first necessary and sufficient conditions for existence of
finite invariant measures, now contained in Hajian's and Kakutani's theorem, were
derived by E.Hopf [1932]. Calderon [1955] and Y.N.Dowker [1955], [1956] gave
progressively stronger results, before Hajian and Kakutani [1964] introduced the simple but
powerful tool of weakly wandering sets. For an approach which avoids Banach limits see
Hajian-Ito [1969]. A streamlined introduction to the special case of nonsingular point
mappings τ can be found in the paper of Jones and Krengel [1974]. They have also
strengthened the result of Hajian-Kakutani for invertible τ:
Theorem 4.12 (Jones-Krengel). If τ is invertible and nonsingular in (Ω, л/, μ) there exists a
set We stf and a sequence 0 = k0 < kx < ... of integers such that the setsx~ki Ware disjoint
and their union is D.
Note that the existence of such an "exhaustive weakly wandering set" is not even simple if
τ is the translation τ: χ -► χ + 1 in Ζ.
The^same paper also contains other conditions; e.g., if $0 is countably generated, we
have С = 0 iff there exists an A with an orbit {τ* А, к ^ 0} which is dense in s/ with respect
to the metric d(A, B) := μ(ΑΑΒ). (We assume μ(Ω) = 1.) Kamae [1983] has characterized
the set of weakly wandering sequences; see also Ellis-Friedman [1978], [1978a].
n+j-l
By Corollary 4.7 there exists an invariant ν ~ μ iff lim sup η _1 Σ μ (τ "l A) > 0 for all
η j i = j
A with μ(Α) > 0, as was shown by Sucheston [1964]. It therefore is natural to ask if even
the condition
(4.11) lim sup μ(τ~ιΑ) > 0 for all A with μ(Α) > 0
i
is sufficient for the existence of finite invariant measures. A transformation with this
property is called of positive type. Hajian and Kakutani [1964] observed that (4.11) is not
sufficient; a construction of ergodic transformations of positive type with an infinite and

Existence of finite invariant measures 145
hence no finite invariant measure can be found in the paper of Osikawa and Hamachi
[1971]. Hajian and Ito [1978] have prepared a survey on transformations without finite
invariant measure.
Conditions for the existence of finite invariant measures for positive contractions in L1
(in the form of stochastic kernels) were first studied by Y. Ito [1964]. This paper contained
the conditions (Ix) and (I2) in Corollary 4.7. Condition (DSN) has been obtained
independently by Dean-Sucheston [1966], and Neveu [1967]. Neveu also constructed weakly
wandering functions h with μ(Ζ>\ {/* > 0}) < ε. Foguel [1969] showed, by a much longer
argument, that (F) holds for weakly wandering functions 1A. Takahashi [1971]
constructed a strictly positive weakly wandering h in the case Ω = D. Condition (B) and the
concept of га-weakly wandering functions as well as the first half of lemma 4.4 are taken
from the thesis of Brunei [1966]. Theorem 4.6, obtained jointly with Brunei, clearlypwes
a lot to the above papers, primarily to Neveu, who stated the decomposition Ω = Cu /5
with a different characterization.
Its purpose is to unify and strengthen the above conditions and various other conditions
of Neveu, Y. Ito [1964], Hajian and Ito [1965], [1967], Krengel [1967], and S. Horowitz
[1968]; see also Helmberg [1966a]. [Apparently, Foguel [1969] overlooked that the
decomposition, even in Neveu's form, requires the present generality of theorem 4.2 and
not just the case Ω = 2?.]
By theorem 4.6 D can we written as a countable disjoint union of sets X(i) with
lim <μ, A* lX{i)> = 0. One can take X(l) = {h0 ^ 1}, X(n) = {n'1 ^h0<(n- l)'1},
(n ;> 2). This can be used to derive also conditions of a pointwise nature. We leave the
proof of the following elementary lemma as an exercise: If, for each i i> 0, cin,
00
(n = 1, 2,...), is a non negative null sequence and the convergent sums Σ cin tend to some
i = 0
о 0 for η -> oo, there exists some Vcz {0,1,...} for which the sequence Σ» e ν сы diverges.
Assume that the limit of the decreasing sequence <μ, Τ*η 1D> is positive; (e.g., Tis given by
a nonsingular τ with Ω Φ С or C\ С has positive measure). Apply the lemma with cin
= {μ,Α*1Χ(ί)}. Then A = \JieVX(i) has the property that the sequence {μ, Α*1Α}
diverges.
In particular, A* 1A cannot converge a.e. (and not even stochastically). On the other
hand, we know from the previous section that in the case of existence of a finite invariant
measure μ0 ~ μ the sequence A* ^converges a.e. for all A. We find that for conservative Τ
and for Τ given by a nonsingular point mapping τ the a.e.-convergence of A* 1A for all
A e $0 is necessary and sufficient for the existence of a finite invariant μ0 ~ μ. Such
conditions are due to Mr§. Dowker [1951], [1955] for point mappings and to Feldman
[1962] for operators, see also §4.3, and Chersi [1982].
In the case Ω = С the sequence Ani is uniformly integrable by the remark after
definition 3.13. On the other hand, if Ani is uniformly integrable, then the stochastic
convergence of An\ implies that <^4„1,1A} = {μ, A* 1A} converges for all A. Hence a
conservative Tadmits a finite invariant measure μ0 ~ μ iff the sequence An 1 is uniformly integrable;
see Υ Ito [1965], and Kim [1968].
An interesting condition has been proposed by Brunei [1970]:
Theorem 4.13. A positive contraction TinL1 possesses a finite invariant measure μ0 ~ μ iff
00 00
for all a = (л;)^0 w^tn ail=Q and Σ at = 1 the operator Ta— Σ atTl is conservative.
i = 0 i = 0
An application showing that all random walks on a locally compact metrizable group G
are recurrent iff G is compact has been given by Brunei and Revuz [1974]. For
generalizations and related work see Horowitz [1972], Falkowitz [1973], Foguel-Weiss [1973].

146 Positive contractions in Lx
Krengel [1967] has given a different condition involving weights. He shows that С is a
special weighted conservative pari. If w = (w^o is a decreasing sequence of positive num-
00
bers with divergent sum, Cw ·= { Σ wk Tkf0 = oo} is independent of the choice of a strictly
positive integrable/0. С is contained in all sets Cw and, for some н>, С = Cw. Aaronson
[1983] used this to show that the larger the eigenvalue group of a nonsingular
conservative τ is, the less recurrent it is.
D. Maharam [1969] derived rather different necessary and sufficient conditions for the
existence of an equivalent finite invariant measure. They involve the pointwise relative
frequencies of the sequence of «'s for which Τηί (ω) ^ α holds.
The conditions for the existence of a finite invariant measure for a strongly continuous
semigroup {Tt, t ^ 0} of positive contractions in Lx are quite analogous; see Lin [1972]. If
ι
p0 e L\ is 7i-invariant S01p0 = J Tsp0ds is 7^-invariant for all t ^ 0. If С is the strongly
о
conservative part for Tt and p0 is such that {p0 > 0} = С holds, then we must also have
{^o,i Po > 0} = С because => holds by the strong continuity at t = 1 and с because
S0lp0 is 7i-invariant. Thus, the strongly conservative parts agree for all Tt, (t > 0).
The general conditions derived in this section have been used in proofs of some general
theoretical results. They are frequently not very helpful for specific examples which
require methods tailored to their special structure; see, e.g., Lasota-Yorke [1973], Veech
[1982], Cornfeld-Fomin-Sinai [1982].
The existence of finite or σ-finite invariant measures for nonsingular transformations is
closely related to the theory of orbit equivalence developed by Dye, Krieger, and others.
We refer to Weiss [1981] for an introduction, and to Hamachi-Osikawa [1981], Hajian-
Ito-Kakutani [1972], [1974].
For abstract generalizations in Banach lattices of several results of this section, see
Shields [1967], Kuhne [1982a], Akcoglu-Sucheston [1984], and Brunel-Sucheston
[1984/85].
More on invariant measures can be found in § 4.3 (Power bounded T), § 6.3
(semigroups), and in the supplement (σ-finite measures).
§ 3.5 The subadditive ergodic theorem for positive
contractions in Lx
We now prove the extension of Kingman's subadditive ergodic theorem to
positive contractions in L1? due to Akcoglu-Sucheston [1978]. The key step is a
generalization of Kingman's decomposition of a subadditive process into an
additive process and a non negative subadditive process with time constant 0.
A family F= {Fn,n ^ 1} of integrable functions is called a subadditive process
(for a positive contraction T) if the conditions
(i) Fm+nuFm+TmFn, (m9n*i),
(ii) sup ||H-1i^||i < oo
и
are satisfied. If —Fis subadditive Fis called super additive, and if Fis subadditive

The subadditive ergodic theorem for positive contractions in Lx 147
and superadditive Fis called additive. If Γ is the composition with an endomor-
phism τ of (Ω, «я/, μ) this agrees with the definition in § 1.5:
If {Ц k\ (/, к) e Q) is given as in § 1.5 we may put Fn = F0n and obtain the
formulation used here. Conversely, if {Fn: η ^ 1} is given, we may put Fitk = T{Fk-{ to
obtain the formulation with the double index.
For the operator Τ = 0 every decreasing sequence Fn with the property (ii) is a
subadditive process and simple examples show that n~iFn need not converge in
any sense.
Recall that Γ had been called Markovian if T* H =11 holds, or - equivalently -
if \Τ/άμ = \/άμ holds for all/e Lv This is certainly the case for the point
mapping case 7/ = /° τ. If Τ is Markovian, condition (i) yields §Fm+n = \Fn
+ J Fm and lemma 1.5.1 shows that the sequence η -1J Fn converges. We may then
define the time constant as before by
y(F) = ton л-1 {/£<///.
It is convenient to work mainly with superadditive processes (Fn). Then the pro-
n-l
cess F„—Fn— Σ TlF1 is non negative and superadditive and differs from the
i = 0
original process only by an additive process.
n-l
An additive process G = (Gn) necessarily is of the form Gn = Σ Τ* Gx for some
i = 0
Gxe Lv It is called an exact dominant for a superadditive process (Fn) ifGn^Fn
holds for all η = 1 and the time constants agree: y(G) — y(F). Sometimes Gx is
called the exact dominant because it determines (Gn). As Γ is Markovian we have
γ (G) = J Gx άμ. By ( — Fn) = ( — Gn) + {Gn — Fn) the existence of an exact
dominant for a superadditive process (Fn) is equivalent to the existence of a Kingman
decomposition of a subadditive process ( — Fn) into an additive process and a non
negative subadditive process with time constant 0. Therefore the following result
of Akcoglu-Sucheston is a precise generalisation of Kingman's decomposition
theorem to the case of Markovian T:
Theorem 5.1. Let Τ be a Markovian positive contraction in Lu then each
superadditive process (Fn) for Τ has an exact dominant.
The following lemma is due to Kingman in the case of measure preserving
transformations:
Lemma 5.2. Let (Fm) be a non negative superadditive process for a Markovian T.
The sequence
m
<Pm = m~x Σ (Ft - ТЦ_Х), (with F0 = 0)
i = l
of non negative integrable functions has the properties

148 Positive contractions in Lx
(α) \q>^^y{F), (w^l),
(β) Σ* Г'?-£ (l - —V"' (т^1,1йп<т).
j=o \ m J
Proof. Fi^t F1 + TFi_1 implies <pm ^ 0. For superadditive (Fm), y(F) is equal to
sup m'1 §Fmdμ. We therefore obtain
Ш
ΙφΜάμ = m-1 Σ f №~ Τ^.χ)άμ = m"1 Σ ί^-^-^φ
i=l i=l
= lfl"1J/Slrf|l^y(/0.
The inequality (β) is a consequence of the estimates
n-l m-1 n-l
ю Σ ^>w = (/- П Σ ^+ Σ г'/ς
J=0 i=l i=0
n-l m-и
= Σ^ + ^η+ Σ (Fn+i-T"Fj
+ "Σ (Γ'/ζ. - r^.+w_n) £ (1 + (m - n))Fn,
i=i
where the last inequality follows from Ft^0, Fn+i — ТпЦ^ Fn, and
P(Fm+Tn^Fj+m_n)^PFn_j^0. D
If the sequence <pm would converge in Lx the proof of theorem 5.1 would already
be complete because we could take Gx — lim (pm. Unfortunately, the sequence cpm
need not converge and a limit point has to be determined by some sort of
compactness or subsequence argument.
Proof of theorem 5.1. Let us first assume that si is generated by a countable
algebra si0 a si. Let A be a strictly positive integrable function. si0 is dense in si
in the metric d{A, B) = (Α μ) (Α Α Β). For each i ^ 0, j'^ 1, and A e si0 the
sequence (Т1(рт a (jh), iA} is bounded. By the diagonal procedure we may find a
subsequence McN such that
lim <Г>тлОУ0?1л>
meM
exists for all i Ξ> 0J^ 1 and A e si0 and hence also for all Ae si. Write w-limM
for the weak limit in Lx along the subsequence. We have obtained the existence of
elements Ay e L\ with Ay = ц?-Итм(Т*(рт л (JК)). For fixed i the sequence Ay is
non decreasing in j and norm bounded by sup 11 <pm 11 x rg y(F) < oo. Hence
Нт^да Ay = A,- exists a.e. and in L1? and J Α^μ ^ у (F). The inequality (/?) above
implies that if η < m, then
n-l n-l
Σ [(7>J л C/A)] ^ [ Σ 7>J л α*) ^ 1 - - К a QA).
i = 0 i = 0 V ™
η

The subadditive ergodic theorem for positive contractions in Lx 149
Taking the w-limM and then passing with j to oo we obtain
(5.1) ZU4 (n*i).
i = 0
For each у"^ 1 and ε > 0 we can find an integer к and a function ge L^ with
\\g\\1 < ε and T(Jh) й к · h + g. As the continuity of Tin the strong topology of
Lx implies also the continuity in the weak topology we have
Ткц = Т{м-Итм(Г(рт л С/А)} = и;-ИтмГ(7>т л (#))
£ и;-Нтм(Г+>т л (*А)) + g £ А|+1Д + g £ Л|+1 + g.
As j was arbitrary and ε > 0 arbitrarily small, Τλ{ rg Ai+1 follows. Let us show
that
00
Gi = ^0 + Σ (Λ+1 — rAj)
i = 0
has the required properties. All summands are non negative.
Using J TXt = J Я£ and J A{ ^ у CO we can show
JGX = lim J(A0 + £ (Я(+1 - U)) = Ит pn+1 ^ y(f).
и-* 00 i = О n-> oo
We can write
A, = (A, - 7VJ + Г(А,_Х - ГА,_2) + ... + Г"1^ - Τλ0) + ГА0.
With (5.1) this yields
i=0 v = 0
The additive process (Gn) dominates (Fn). Therefore we also have J(?i = y(G)
It remains to eliminate the assumption that si is countably generated. This is
done with a standard argument: If sf1 is the smallest σ-algebra in which Fl9F29...
are measurable then s/1 is countably generated. Once a countably generated sik
has been constructed we know that Lx (Ω, sik, μ) is separable. Let/kl,/k2> · · · be
dense in this space and let s/k+1 be the σ-algebra generated by all Tlfkj, (i ^ 0,7
g; 1). The σ-algebra si ^ generated by the sequence s/u sf2>... is again
countably generated, all Fn are si^ -measurable, and Γ maps Lx (Ω, si^ μ) into itself.
We can therefore apply the above argument in this space, α
It is now not hard to prove a convergence theorem for superadditive processes:
Theorem 5.3 (Akcoglu-Sucheston). Let Τ be a conservative positive contraction
in Lx and let (Fn) be a non negative superadditive process with exact dominant (Gn).
Then limw_aoFJGn = 1 a.e. on {SO0G1 > 0}.

150 Positive contractions in Lx
Proof. We may assume Ω = {So0G1 > 0}. We shall use the following notation: h
= lim infFJGn, h = lim supFJGn, L(/? g) = lim Snf/Sng, Fnk = V Fkiiv As
n-l _ и i = 0
(Gn) is a dominant, Л rg 1. As the sequences (Fn) and (C?„) are increasing in η and
lim Gnk/Gin+1)k = 1 by Lemma 2.3 we see that h = lim inf F(n+i)k/Gnk. The super-
additivity implies TjFnk й ^„+i)fc for; = 0,..., к - 1 and Fnk ^ Fnk. It follows that
fc-1 n-l
k~xSnkFk = AT1 Σ Σ :Tw+'Fk ^ ^„+i)k and hence also that
j = 0 i = 0
jA</(Gi · μ) ^ {ДАТ1/;, GJ </((?! · μ) = AT1 jFk^.
As /c-1 JffcJjU tends to y(F) = \ΰχάμ ^ {/^(G^ - μ) we have proved A = A on
{Gi>0}.
The assumption {C?! > 0} = Ω is no restriction of generality: Simply let C?i
00
= Σ 2~iv+1)TvGl9 GX = Gn + SnG[, F,;' = Fn + SnG[. Then the result just
v = 0
proved yields limF^/G^ = 1. As $1EG[ = J1 £С?х holds for all Ее <€, we have
SnG[/GX -* 1/2 and C?n'/Gn ^ 2 on {S^ > 0} = Ω, and hence FJGn -* 1. α
One can also obtain convergence a.e. on С when Г is not conservative by
observing that for non negative superadditive (Fn) the restriction (lci£) is super additive.
For non negative subadditive (Fn) the sequence Fn converges a.e. in D because of
00 00
Σ (Fn+1 — Fn)+S Σ TnFi < oo. One can also study ratios FJHn where Hn is non
n=\ n = l
negative and satisfies Hm+n ^ Hm + TmHn, but the boundedness assumption
sup \\rn~1 Hm\\i<co fails. This is analogous to Chacon's admissible sequences in
the previous section and generalizes them.
Notes
It seems that at present all proofs of the existence of the Kingman decomposition depend
on variants of lemma 5.2 and differ only in the limiting argument. Kingman used w*-
compactness. Burkholder [1973] gave a short proof, which - however - depends on the
deep theorem of Komlos [1967] that each Ll -bounded sequence admits a subsequence for
which the Cesaro averages converge a.e.. Del Junco [1977] gave a martingale argument.
The present argument of Akcoglu-Sucheston has the advantage of being both short and
elementary. Brunei and Sucheston [1979] have studied the existence of dominants for non
negative processes with Fn+m ^ Fn + TnFm, (m, η ^ 1), when the positive contraction Τ
need not be Markovian. Call an additive process (Gn) a dominant if Gn ^ Fn (n ^ 1) and
exact if J Gn is minimal among all dominants. They show that the sequence J cpm converges
to some limit α with 0 5Ξ α ^ oo and an exact dominant exists iff α is finite. Then α
= lim J(pm. Akcoglu and Sucheston [1982] have a superadditive version of the lemma of
Brunei for Ecz C. Fong [1979] has studied ratio and stochastic superadditive ergodic
theorems for positive Cesaro bounded linear operators Γ in Lx.

An example with divergence 151
Brooks and Chacon [1980] derived the existence of exact dominants from a general
lemma on additive set functions. Akcoglu-Sucheston [1984b] considered processes Fn
= \p(Snf) with/^ 0 and subadditive or superadditive functions ψ on IR+.
§ 3.6 An example with divergence of Cesaro averages
We shall now consider Chacon's [1964a] example of a positive conservative and
ergodic contraction Tin Lu for which the averages ^„/diverge a.e. for all non
zero/e L\. Twill be induced by an invertible nonsingular point mapping τ: for
ν g Lx we shall have Tv — ν ° τ-1. Γ* has the form Τ*h — h ° τ. As the averages
^„/converge a.e. in the case of existence of a σ-finite invariant measure μ0 ~ μ,
the example also shows that such a μ0 need not exist in general.
It will be enough to show the existence of one element fe L\ for which Anf
diverges a.e., because then the divergence for all other Оф/'е^ follows from
the Chacon-Ornstein theorem. The/in the construction shall even satisfy
(6.1) liminfylw/= 0 a.e., and
(6.2) limsupn~iTn~1f=oo a.e..
Ω shall be (mod nullsets) a subinterval [0, a] of IR+ endowed with the Lebesgue
measure μ. The value of a > 1 shall be determined in the construction. If /
= IX β I and / = [y, δ [ with oc< β and у < δ are two intervals the affine
transformation from I to J shall be the map
τ(ω) = 7 + -—£(ω-α).
τ induces a positive contraction Τ from Lx (I) to Lx (J): For any ge Ьх(1) we
have (Tg) (η) = (β — α) (δ — y)-1/(T_1 ή) f°r Ц е J- This is just the description of
the map ν -* Tv = ν ° τ-1 from Lx (I) to Lx (J) in terms of Lx (/) and Lx (J). (We
shall use the letters τ, Talso for the restrictions of the ultimate transformations to
subsets).
τ on Ω shall be defined by an inductive procedure. We start with /{ = [0,1/2 [,
I\ = [1/2,1 [, Nx = 2, and τ on l\ shall be the affine map I\ -► 1\, At the л-th
stage of the construction we have defined a finite number Nn of left-closed, right-
open disjoint intervals /J,..., 7£n such that their union Ωη is an interval [0, an [.
The sequence Ωη shall be increasing with union Ω. τ on Ц is the affine map to Ц+и
(ν = 1,..., Nn — 1). Set/= 1[ο,ι/2[· To simplify the notation we write TV for 7V„, N'
for Nn+l9 Iv for Ц and /rv for Ц+1 in the inductive step.
Passage n -* л + 1 /or л яяИ. We take /^ = /v for ν = 1,..., N and we add new
disjoint intervals Γν9 (ν = Ν + 1,..., Λ^). The number 7Vr will be specified in a
moment. The new intervals shall form a subdivision of [a„, an+x [ with απ+1 = an

152 Positive contractions in Lx
+ μ(7ι)/2. If Mn is any integer with Mn ^ N + 1 and TV' = 2Mn we can be sure
that (Akf) (ω) is well defined at this stage of the construction for all к ^ Mn and
ω e Ωη: If ω belongs to Ip (1 uju Ю, then (Pf) (ω) is determined for 0 S i uj
— 1 by the sizes of the intervals Ij9 Ij_l9 ..., ^andby/iT^co). For/?g /<M„we
can already be sure that τ-ίω shall not belong to [0,1/2 [, no matter how we
continue the construction. For these i we shall therefore necessarily have
(Tlf) (ω) = 0. This shows that by a sufficiently large choice of Mn we can achieve
(6.3) 0uAMnf(co)<l/n, (ωβΩη).
We complete this step of the construction by fixing such an Mn. τ is defined on
Ωη+ι \Γν* by the requirement that τ maps Γν -* Γν+ΐ9 (ν = 1,..., Ν' — 1) in an
affine way.
Passage η -+ n + 1 for η even. Now the construction depends on an integer К
K-l
= Kn ^ 2 and positive numbers ank, (0 ^ к < K), with Σ <xnk = 1, to be deter-
k = 0
mined in a moment. Each of the intervals It is cut into AT disjoint subintervals Iik
of length ocnk ju(/f) such that, for A: = 0,..., K— 2, the interval Iitk+1 lies to the
right of Iik. The restriction of τ to Iik automatically is the affine map to Ii+lfk.
Set N' = N · K. Each j with 1 uju N' has a unique representation of the
form ./=£#+ι with 0^к<К and 1 ^ ι й N. Define Τ; = Iuk. On the
set Ωη\ΙΝ where τ has already been defined previously we just observed that τ
restricted to Ij is the affine map to Ij+1. In order to achieve this also for all other
/ < N' we define τ on I'mN — INm_ι as the affine map to Гт^+± — Am?
(m = 1, ..., Κ — 1). We arrive at the desired situation τ: /i -* /2 -* ... -* ^лг ·
We now determine the parameters К = Kn and ank: Set επ = 2~n. There exists a
constant cw = о 0 and an integer К ^ 2 with
(6.4) sn>cnN and 0 < 1 - (εΒ + с Σ (^ + Ι)"1) < £„·
k = l
Define αη0 = β„, аик = с (Л + I)"1, (А: = 1,..., К- 2)
К-2
ап,К-1 — 1 — Σ апк·
к = 0
By (6.4) we have ocnk S απ0? (k = 1,..., К — 1). As this is true for each stage of
the construction and the added intervals for the steps with η odd were shorter
than the first interval Ιγ of that step, we can be certain that in each step the first
interval in the chain Il9..., IN is the longest. It is also always a subinterval of
[0,l/2[.
The construction is complete. As the measure of the set I^n υ (Ω \ Ωη) where τ is
not yet defined at stage η tends to 0, τ shall ultimately be defined on almost all of
Ω, and similarly also τ"1. After the elimination of a nullset, τ and τ-1 are defined
everywhere. (6.1) is a consequence of (6.3). Let us now check (6.2):

An example with divergence 153
For η even, let Bn be the set of points ω belonging to some Ij = I?+1 with N + 1
Sju(K-l)N. We may write j= kN + i with 0<k^K-2 and 1 й i й N.
г7"1 maps I[ onto Ij. We therefore have
Kb) Щ)
μ(ΐ]) c(k + \)
Using μ(Ωη\Βη) ^ 2ε„μ(Ωπ), we see that almost all ω belong for infinitely
many even η to Bn. This implies (6.2). In particular, Γ must be conservative.
It remains to show that Τ is ergodic. Let Ε be a τ-invariant set of positive
measure. Let s/n be the algebra generated by /?, ...,/^n? and s/^ = \JsJn. As the
length of the longest interval T[ of step η tends to 0, s/^ generates the Borel-σ-
algebra si in Ω. Hence, for any η > 0, we can find an л and a set i^ e J3/W with
Fn с Ω„ and μ(^ η Я) ^ (1 - η) μ{Ε).
As i£ can be chosen arbitrarily close to Ε we can also attain μ(ΕηηΕ)
^ (1 — 2η)μ(Εη). But then there must be at least one of the intervals Ц with
μ{ΓνηΕ)^{\-2η)μ{ΓΙ).
As Ε is invariant and τ affine we must then have μ (Ц n E) ^ (1 — 2η) μ (Ц) for
all ν with 1 ^v ^ Nn. This implies μ(Ε) ^ (1 — 2η)μ(Ωη). As η was arbitrarily
small and n arbitrarily large we obtain μ(Ε) = μ (Ω), α
Notes
The first example of a nonsingular transformation τ admitting no σ-finite invariant μ0 ~ μ
was given by Ornstein [1960]. The construction of Chacon is partly similar to Ornstein's
example. Such constructions are now called stacking constructions, and they are discussed
in some detail in the book of Friedman [1970]. Also the simple example of Brunei [1966a]
belongs in this class. Hamachi [1981] gave a very different example: a Bernoulli shift with
nonidentical factor measures.
It is simple to see that there exist positive linear operators in L1 with || Γ|| <Ξ 1 + ε and
II ^ΊΙι,οο = 1 f°r which v4„/need not converge а.е..
Modifying the example above, Friedman [1966] has proved that also ||Г||^1,||Г||1оо
5Ξ1 + ε is not sufficient. A. Ionescu-Tulcea [1965] has proved a related category theorem:
In the class of positive invertible isometries of ^([O, 1]) the set of Τ for which Anf
converges a.e. for all/e Lx is a set of first category in the strong operator topology.
Weiner [1968] showed that there exists a nonsingular τ without σ-finite invariant
measure equivalent to μ, such that An( T)/converges a.e. for all/, where 74s the
corresponding contraction in L1. Y. Ito [1981] has simplified and generalized the construction.
Positive results can be proved using stronger summation methods; see §8.2.

154 Positive contractions
§ 3.7 More on the filling scheme
As the filling scheme has been the key for the proof of the Chacon-Ornstein
theorem we explore it and its consequences a bit further. We also report briefly on
abstract generalizations of the ratio theorem. Except for the first lemma below
which will be applied in § 7.1, the results of this section will not be needed in the
sequel.
We again start with the general situation, where 74s an arbitrary positive linear
operator in a general vector lattice L of measurable functions on Ω. Again
Uh := Th+ — h~. Given/, g e L+ we shall be interested in the sequences hn — Unh,
(n ^ 0), with h = /— g.fn ··= h+n is the matter and gn := h~ the antimatter (or hole)
remaining at time n. The portion dn of the hole filled exactly at time η is given by
d0 =/л g = g — g0, and dn — gn^1 — gn, {n ^ 1). The following lemma
essentially is the original formulation of the filling scheme:
Lemma 7.1. For alln^O, Tnf = fn+ Σ Tn~kdk. On the set В = {Mn+1h>0}
k = 0
η
one has gn = 0, or (equivalently) g = Σ dk.
fc = 0
Proof. For η = 0 the first assertion follows from/= (/— g)+ + (f л g) and the
induction hypothesis yields
Tn+1f= Tfn+ Σ Tin+i)~kdk
fc = 0
= (Tf„-gn)+-(Tfn-gtt)-+gn+ Σ r<"+1>-4
fc = 0
=fn+i-gn+i + gn+ Σ γ<"+1>"4
fc = 0
fc = 0
The second assertion is equivalent to theorem 2.2. α
The effect of the filling scheme may be more visible if one considers the sums of
Tf9 (и = 0,..., N) and of Pg = T>(gN.j + d0 + ... + dN.j), (j й Ν). Then
Ν Ν
(7.1) SN+1f= Σ/» + ^ι and SN+1g= Σ TJgN-j + uN+1
n=0 j=0
with
"N+i = (do + -·· + dN) + T(d0 + u.. + dN-1) + ...+ TNd0 ^ 0.
In this form the scheme has been used by Neveu [1979] to give a transparent
alternative proof of the Chacon-Ornstein theorem including the identification of

More on the filling scheme 155
the limit in the conservative ergodic case. (It seems simpler to derive the general
case directly than to deduce it from the ergodic case because the ergodic
decomposition of Γ, due to Jacobs [1962/63], (see also Krengel [1963]), is non trivial).
In the case considered by Neveu, the assumption g^ --= limgw φ 0 (that the hole
is not filled completely) implies ψΑ = H in the following lemma:
Lemma 7.2. Let Τ be a positive contraction in Lv Set A — {g^ > 0}, then
(7.2) ΣΛ<οο on{Vil>0}
n = 0
and
(7.3) limJ/BV^ = 0.
η
00
Proof. Set vp = (IAC T*)p \A. Then xpA = Σ vp. Because of/„ л gn = h+n л h~ = 0
p = 0
and gn ^ g^ the functions/,, must vanish on A. Hence fn+i rg Tfn = TIAcfn> and,
by induction, fn ^ (TIAC)nf0. This yields
(7.4) </„, vp} й <(ТУ"/0, (1АсТ*У1АУ = </0, vp+n}.
00
Summing over/? gives </„, ψΑ} rg </0, Σ vj) -* 0, i.e., (7.3). Summing over я
gives <ΣπΛ? ^p> ^ </o, Σπ νρ+η> й <foJ> < oo, so that ΣΛ converges on
{vp > 0} for each ρ ^ 0. α
Rost [1971] has studied the following two questions on the filling scheme for a
positive contraction Tin Lx:
(a) When will the hole be filled completely?
(/?) When will all the matter fall into the hole or disappear?
In other words: When do we have g^ = 0, resp. lim ||/„||i = 0? We shall need
the partial order in Lx defined by h \~ h" iff <A', и) ^ <A", и) holds for all super-
harmonic ue L^.
For f=r + s, (r,se L+x), and f' = Tr + s, we clearly have </, u} = <r, u}
+ <s, u} ^ <r, Γ* u} + <5, w> = </', w>. We therefore see that/^/implies/h):
Similarly, for any de Lx the inequalities <J+ — d~, u) ^ <J+, Г*и> — <яГ, w>
= <7W+ - <T, w> = <£/</, u} give J h C/</. As <Un(f- g)9 u}
decreases, we see </- g, w> ^ lim (Un{f- g), u} = lim </„ - g„, w>
^ lim < -gB, u) = - <&,, w>, i.e.,
(7.5) /-ih-fc.
In particular, /h g is obviously necessary for g^ = 0.
Theorem 7.3 (Rost). /и the filling scheme withf ge ϋγ the relation f\~ g is
necessary and sufficient for g^ = 0.

156 Positive contractions in Lx
Proof. Let us first show, for de Lu that d h 0 implies Ud Ь 0. Set В = {яГ > 0}.
For superharmonic и define ν = ulB + (T*u)lBc. Then T^u^v^u shows
Γ*ϋ ^ r*w ^ ν so that ι? is superharmonic. It follows that <d, ν} ^ 0. Using
the definition of ν and {d+>0}cuBc we find (Ud,u} = (Td+ - d~,u>
= <d+, T*u) — <J~, w> = <d+, t;> — <яГ, t;> ^ 0. As the superharmonic и was
arbitrary we have proved Ud\- 0. By induction Und\- 0 for all л.
Now assume /hg and set A = {gao>0}. We have just proved (gn,u}
= (fn> иУ f°r aU superharmonic w. As tpA is superharmonic, <gn, т/^>
= </„, Vii)· By (7-3) the right hand side tends to 0 and we obtain <&,,, ψΑ} = 0.
Now ψΑ^ίΑ yields goo = 0. π
As each step in the filling procedure is a partial application of the remaining
matter the theorem tells us that, for/, ge Uu the relation/h g is equivalent to
the existence of a sequence of measurable functions af: Ω -* [0,1] with g = axf
+ a2(T(i-ai)f) + a3(T(i-a2)(T(i-a1)f)) + ....
The answer to question (/?) is quite symmetric: As in Meyer [1969/70] we write И
=| A" iff <A', w> rg <A", u) holds for all subharmonic u.
Theorem 7.4 (Rost). 7л the filling scheme with fgeUuf—\g is necessary and
sufficient for Ц/Ji ->0.
Proof For subharmonic w the inequalities {Ud, u) = <7W+, w> — <d~, w>
= <J+, Г*и> — <яГ, w> ^ <d, w>, valid for any de L1? show that the sequence
(fn — gn>uy = (Un(f— g)>u) increases. If Ц/Jli tends to 0 we obtain
0 ^ lim < — gn, u) = lim </„ — gw, w> ^ </— g, w> for all such и and hence/ =| g.
Now assume/=| g. Exactly the same argument as in the previous proof shows
that d =| 0 implies Ud =| 0. Hence/, =|g„ holds for all n. Let e denote the
(decreasing) limit of T*ni. Then we have T*e = e, and (Tnf 1 -e>
= </, Г*"! - e} tends to 0. Because of fn й Tnf we obtain </„, 11 - e> -* 0.
It remains to show </„, e> -* 0. Set A = {gao>0}. As tpA is superharmonic
T*(e — ipA)+ ^ T*(e — ψΑ) ^e — ψΑ shows that (e — ψΑ)+ is subharmonic.
Now fn =| £w yields </w, e> ^ </w, xpA} + </„, (г - ψΑ) + } й </„, tpA>
+ <£«? (^ "~ ^л) + )· The first term tends to 0 by (7.3) and the second because gn
decreases to 0 in Ac and (e — ψΑ)+ vanishes in Α. α
For further related results and the interpretation of g (in the case g^ = 0) as the
distribution of a stopped Markov process the reader should consult the papers of
Rost and Meyer.
Notes
Let Γ be a positive linear operator in a vector lattice L of measurable functions on
(Ω, «я/, μ). Ornstein [1970] calls fe L+ a modification of/e L+ if there exists some и е L+

More on the filling scheme 157
with/ = /— и + Tu. E.g., for/ = r + s,(r,se L+), the function/^ Tr + s is a
modification of/because we may take w = r. If/' =/— w + 7м is a modification of/and/'' =/'
— u' + TV a modification of/', then/" =/— (w + u') + Г(и + w') is also a modification
of/ We thus see that/-5»/implies that/is a modification of/. (However, a modification/of
_ _ °°
/need not satisfy/-5»/for any «. E.g., we may have/= Σ 2~("+1) Γ"/.)
л = 0
Before we continue let us give an application of modifications:
Theorem 7.5 (Fong-Sucheston [1973]). If Τ is a conservative positive contraction in Ll the
subspace L0 = {he Lx\ <A, 1£> = 0 for all ЕеЩ is the closure of {u — Tw.ue L\}.
Proof For any иe Uu we have u— Tue L0 since Εe <# is equivalent to T* 1E = \E. For
A e L0 consider the filling scheme for/= A+, g = A". As Τ is conservative, the superhar-
monic functions are harmonic and ^-measurable. Hence/1- g. For ε > 0 we may find an η
with ||#Β||ι<ε. Recall gn = g — (d0 + ... + t/n). As Γ is Markovian we have WfWx
= ΙΙ/ι+ιΙΙι + ll^lli- we obtain ii/ju - ц/iu - Σ ll^lli = \\g\\i - Σ ll^-lli = ll^lli < β.
i = 0 i = 0
_ π
/= Σ di +fn is a modification/— u+Tu of/ A =/— g = u— Tu — gn +/„ now gives
i = 0
||A-(w- Ум)||1 < 2e. α
If Tis a positive contraction in Lx a modification/satisfies WfWx = ||/— w + Tu\\1 = \\f\\i
+ \\Tu\\i — \\u\\i^\\f\\i. Therefore the following theorem of Ornstein contains the
Chacon-Ornstein theorem:
Theorem 7.6. Let Τ be a positive linear operator in a vector lattice L of measurable
functions on (Ω, д/, μ), andf g e L+. If Snf/Sngfails to converge on a subset Ε of{Sca g > 0} there
exists a sequence en of modifications off+ g with lim e„(co) = oo a.e. on E.
Ornstein also gives a partial converse. Under suitable assumptions, satisfied e.g. for
positive contractions in L2, the set where the ratio convergence fails for some/, g agrees mod μ
with the set where there exist arbitrarily large modifications. These results and those in the
work of Fong and Lin [1976] show that under fairly general conditions the ratios Snf/Sng
converge a.e. for all/, g e L+ only if 74s a positive contraction in Lx after a suitable change
of the underlying measure.
Foguel [1980] has a Hopf decomposition in vector lattices.
Brunei [1976] has extended many of the results of § 3 of this chapter to the abstract
situation studied by Ornstein. He calls Ε a good set if there exists a non negative χ ^ Τ* χ
in the space L* of bounded linear forms on L which is strictly positive on E. (This means
that there is a ψ e L+ with Ε cz {φ > 0} and χ(/) ^ \φ/άμ V/e ΖΛ)
The condition that there exists no sequence en of modifications of и e L+ with en -» oo
a.e. on is is shown to be equivalent to the condition that Ε is good when L is replaced by the
space Lu of all functions bounded by finite linear combinations of functions Tlu, (i ^ 0).
η
There is also an abstract form of Chacon's convergence theorem for ratios Snf/ Σ Pi with
i = 0
00
(Pi) admissible (asserting convergence on good sets in { Σ Pi > 0}.), and of Brunei's
i = 0
lemma. Gologan [1979] has dropped the condition of positivity of Γ for suitable spaces.
In these generalizations the basic setup is still defined in terms of a linear operator. As the
convergence of ratios qn = (/0 +/i + ... +fn-i)/(Po + · · · + Pn-i) depends only on the
joint distribution of (fo,Po>fuPu ···) Dinges [1971] has proposed to investigate con-

158 Positive contractions in Lx
ditions for convergence which should be stated in terms of these distributions. Let Pn
denote the joint distribution of (fn,pn,fn+i,pn+i, ·.·)· It was shown that the condition
<^ J h dPn ^ J h dPn+1 for all non negative, positively homogeneous convex functions h and
all η ^ 0 ρ could be used to prove convergence a.e. of qn. However, it was shown by Rost
[1971a] and Engmann [1976] that under such a condition qn did arise from a positive
contraction. Engmann also discussed the ergodic theorem of Cuculescu and Foias [1966],
which deals with a sequence 7\ ^ T2 ^ ... of positive contractions and functions fn,pn
with Tn+lfn^fn+l, Tn+l\pn\^pn+1. (See also Gologan [1977]). In the course of his
program Dinges [1970] systematically studied inequalities and identities between partial
k-l
sums 4 = Σ fj of functions and various maxima and minima defined in terms of some 4·
j = i
The book by Dellacherie-Meyer [1983] and the articles by Dinges [1974] and Revuz
[1978] are further references related to the filling scheme.

Chapter 4: Extensions of the Lx -theory
Several results from the previous chapter are now generalized in three different
directions. First we drop the positivity assumption. Then we allow that the
functions take values in W or even in a Banaeh space. Finally, we show that the case of
power bounded or Cesaro bounded operators can often be reduced to the
contraction case.
§4.1 Non positive contractions in Lj
1. The linear modulus of T. Quite a few ergodic theorems have first been derived
for positive operators in L1? but can be extended to the case where Γ need not be
positive with the help of the "linear modulus" of T. We now discuss this device
and some applications.
For positive operators it has not been a restriction of generality to work always
with real Z^-spaces, since a complex/can be decomposed into its real and
imaginary parts. In this section we shall always work with complex Lx-spaces. Some of
the interesting non positive operators like Tf(co) = 6ία(ω)/(τω), (for α ^ 0 and
measure preserving τ), make sense only in the complex Lx. The results will remain
true for operators in the real Lv In fact, in that case some statements can be
strengthened or are simpler due to the lattice structure of the real Lx. Τ in the
following theorem is called the linear modulus of T. It is frequently denoted by
\T\.
Theorem 1.1. For every bounded linear operator T: Lx -> Lx there exists a unique
bounded linear positive operator T: Lx -> Lx such that
<%Ч|тц = |1Л1,
(ii) forallfeLl9\Tf\uT\f\,
(Hi) for allfe L\, T/= sup{| Tg\: ge Ll9 \g\ £/}.
Proof. Let^ denote the family of all finite measurable partitions
π = {Д, jB2? ..., Bm} of Ω. 0> is partially ordered in the usual way: π ^ π' means
that %' is a refinement of π, i.e., the sets Д. are unions of sets of π'. For/e L\
m
define Ύπ/:=Σ\Τ(1Βιη\- Clearly π й π' implies TKfu Τ,./. From Ц/Ц,
i = l
m
= Σ HI,/»! we obtain НТЛ ^ ||T|| Ц/IK. As {Τπ/: яе ^}isincreasingon#
ί = 1

160 Extensions of the Lx -theory
and norm bounded we may define
(1.1) T/.= limT„/, {feL\).
(The monotone convergence theorem for families {/„} a L\ indexed by elements
a directed set 0* follows easily from its sequence version. If πχ ^ π2 ^ ... is an
increasing sequence with ||/π„||ι-► a — sup^l/J^: πβ ^} < oo, then
/πι rg /Я2 rg · · · converges to some/^ e L\ with norm a, and it is easy to check that
/oo = Нт/Я does not depend on the sequence (π„).)
We clearly have
(1.2) ЦТ/ПРИГНИ/!!,, (feL\)
and T(/+ g) й Т/+ Tg, (/ g g Z^). If/and g are simple functions, (i.e., take
only finitely many values), and π is at least as fine as the partition generated by the
sets of constancy of/and g, we have Τπ(/+ g) = Τπ/+ Τπ#, and, in the limit,
T(/+ g) = T/+ Tg. By approximation this remains true for all/, ge L\ and Τ
acts as a bounded positive linear operator on L\. Now Τ can be extended by
linearity to all of Lv This extension is again denoted by T.
For feL\ and |g| ^/we obtain T/^ \Tg\ by approximation with simple
functions. This yields (ii) and the inequality ^ in (iii). If ^ fails in (iii) there exists
an/e L\, a partition π = {Bu ..., Bm}, a set yl0 with μ(Α0) > 0 and an ε > 0 such
that
(1.3) Τπ/^ sup \Tg\+e on Λ-
Now there exists a complex number ax with |ax | = 1 and a set At с yl0 with
μ^) > 0 such that \ax T(lBlf) — \ T(iBlf)\\ rg s/2m on ^χ. Continuing in this
way we find at with \at\ = 1, (i = 1,..., m), and yl0 =5 ^ id yl2 ... id Am with
μ(Λ„) > 0 such that \а{Т{\в/) - \ T{\B.f)\\ <, s/2m on At. Using the function
m
g = Σ a{f\Bo a contradiction to (1.3) results. The identity (i) follows from (1.2)
i = \
and (ii). α
The proof of the following proposition is straightforward and is left as an
exercise:
Proposition 1.2. The mapping Τ -> | Г| ш the previous theorem has the following
properties:
(a) T^O implies T=\T\;
(b) |аГ| = |а||Г|;|71 + Г2|^|7Ц + |Г2|;
(c) \Т,Т2\ й\Тх\\Т21{< occurs)'
(d)\T\n\f\^\Tnf\ forallfeLun*0;
Г^ II ΤΊΙι.οο = III 3^1 lli.oo; (see § 1.6);

Non positive contractions 161
(f) The map Τ'-> | 7Ί is norm continuous. (In general it is not continuous in the
strong operator topology.)
The arguments above can be modified to show that the space of bounded linear
operators in a realLx is a lattice. E.g. T+ = Tv 0 and Γ = (-Γ)νΟ may be
constructed by setting (for / ^ 0)
m m
τπ+/= Σ (П1в/))+, т-/= Σ (Γ(ΐΒ|/)Γ;
i=l ί=1
Γ+/=ΙίιηΤ+/ and Г-/=ШпТ-/.
It is then simple to check \T\ = T+ -T~.
The results above are due to Chacon and Krengel [1964]. Dunford-Schwartz
[1956] gave a proof of the special case of theorem 1.1 in which Τ satisfies the
additional assumption ||7T||lt00<oo. However, in the real case Kantorovic
[1940] has already given a (different) construction of the linear modulus in a
lattice theoretic context.
Remark. Call a linear operator Tin Lp with 1 ^ ρ ^ oo majorizable or regular if
there exists a bounded linear operator T0 in Lp with | Tf\ rg T0 \f\ for all/е Lp.
If Г is majorizable essentially the same proof yields the existence of a linear
modulus Τ satisfying (ii) and (iii); while (i) is replaced by ||T|| ^ || Γ0||.
Theorem 1.3. All bounded linear operators R in L^ of a σ-finite space admit a
linear modulus R. // satisfies ||R|| = ||i?|| and
(1.2) R/=sup{|*g|:|g|£/} forfeLt
If Τ is a bounded linear operator in Lx and R = T*, then R = T*.
(Except for the easy inequality \Rg\ ^ T*|g|, this theorem, due to Krengel
[1963], will be used only in the supplement.)
Proof. Define R7r in analogy to Τπ. A priori the almost sure limit R/= lim R^
might assume the value oo, but the second part of the proof of theorem 1.1
remains valid and shows (1.4). Hence the existence of the linear modulus and
||R|| = ||/?|| follows.
For any he L\ we have
|<A,i?g>| = |<77*,g>| й <TA, |g|> = <A?T*|g|>.
Hence \Rg\ й T*|g|, and R ^ T*.
To prove ^ we use the fact that the natural embedding к: L^-* L** satisfies
K(hx ν h2) = (κ/*ι) ν (κΑ2); see Dunford-Schwartz [1958: 303]. Here hl9 h2 are
real valued, but \к!г\ — к\к\ holds for complex valued h e Lx by approximation
with simple functions.

162 Extensions of the Lx -theory
For/e L+, he L\ we obtain
|</, κΓΑ>| = |</, Th}\ = |<Л/, A>| g <|Д/|, А>
^<R/,A> = </,R*kA>.
As /eL* was arbitrary, к|ГА| = [кГА| ^ R*kA. For π = {Bl9 ..., 2?m} put Af
= A1B|. Then
ic| TAf| й R*Kht й Т**кА* = icTA,.
Summation yields κΤπΑ = R*kA^T**k:A = kTA, and a passage to the limit
implies κΤΑ ^ R*kA ^ Τ**κΑ = κΤΑ. Hence equality holds. Now
<A,T*/> = <Τ**κΑ,/> = <R*kA,/> = <kA,R/> = <A,R/>
shows R = Τ*, α
Theorem 1.1 can be used to prove the ergodic theorem of Dunford-Schwartz
[1956] which asserts that ^„/converges a.e. for all/e Lx if 74s an arbitrary Lx
— L^-contraction which need not be positive.
For economical reasons we proceed right away to the proof of the deeper ratio
theorem of Chacon via a representation theorem of Akcoglu and Brunei [1971].
2. The structure of Ton the conservative part of | T\. In the sequel Twill be a fixed
positive contraction in Li9 Τ its linear modulus, D and С the dissipative and
conservative part of Ω for the operator Τ and ^ the σ-algebra (in C) of T-
absorbing subsets of С It is not difficult to check that a set A e si is T-absorbing
iff Τ maps Lx (A) = {fe Lx: {/фО}с A} into Lx (A).
Lemma 1.4. If he L^ and T^h — h then T* | A | = | A | on C.
Proof. | Г*| = T* yields |A| = |T*A| ^ T*|A|. Apply theorem 3.1.3. α
Lemma 1.5. Assume АеЧ! and A e L^. Then T* A = A holds on A iff\Ah has a (€-
measurable restriction to C.
Proof This is essentially a rewording of lemma 3.3.3. Because of T*l^c
^ 1 Ac, T* A in A does not depend on the values A takes in Ac and we may assume
Ω = A = С Now it does not matter that A ^ 0 has not been assumed, because we
may add a multiple of H = T*H to A to achieve a reduction to the case A ^ 0. D
Lemma 1.6. Let Ae^ and heL^. Then (i): Τ (A/) = A · Tf holds for all
fe LX{A), is equivalent to (ii): Ύ* h = h on A.
Proof Τ is Markovian on A a С Hence hfe Lx (A) and (i) yield </, A> = J A/
= JTC/Ά) = J (A -I/) = <T/, A> = </,T*A>. As fe L,{A) was arbitrary (ii)
follows.

Non positive contractions 163
Now let Ж be the class of all he L^ satisfying (i). If В e %? is contained in A and
fe LM\ then T{\Bf) = 1ВТ(1*Я = U^if- ^л\вЛ) = UU which means
that lB belongs to Ж. Since Ж is a closed linear manifold in L^ containing all A
with h\A = 0 we see that Ж contains all A satisfying (ii). α
Lemma 1.7. Iff g are integrable, \f\ rg g and ^(g ~Γ)άμ = 0, thenf= g a.e..
Proof Since Re{g -f) = g-Re(f)^0 and Re($(g -/)) = 0, it follows that
g = Re(f) and hence that g = /. α
The next lemma will give us the key for the representation.
Lemma 1.8. Assume that Τ is conservative and that there exists an he L^ with
Α φ 0 a.e. and T*h = h. Let s = h/\h\. Then T/= sT(sf)for anyfe Ll9 wAere5
= ί/s is the complex conjugate of s.
Proof It is enough to prove the lemma for/^ 0. Using lemma 1.4 we see that
T*|A| = |A|. Now ί(ΤΛ·|Α| = </,ΤΊΑ|> = </,|Α|> = ί(ίΟΑ = ί(ί/)Γ*Α
= $(T(sf))h = $(sT(sf))\h\ and |*7W)| |A| = | T(sf)\ |A| ^ CW|) |A|
= (T/)|A|? together with the previous lemma yield (sT(sf))\h\ = (T/)|A|.
Because of Α φ 0 a.e. the proof is complete. D
Applying this lemma to the restriction of Τ to absorbing subsets of С we obtain:
Lemma 1.9. Let Ge4> and assume that there exists an he L^ with Α φ 0 a.e. on
G and T^h — h on G. Let s be any function such that s = h/\h\ on G. Then
Tf=sT{sf)forallfeL1(G).
Proof TLX (G) a Lx (G) and | Tg\ й Τ|g| shows TLX (G) a Lx (G). If T', Г, and
A' are the restrictions of Τ, Γ, A to G, then it is simple to check that T' is the linear
modulus of Τ and T', 7", A' satisfy the assumptions of the previous lemma, α
We are finally ready to prove
Theorem 1.10 (Akcoglu-Brunel). Let Τ be a contraction in L1? Τ its linear
modulus, С the conservative part of Ω for Τ an d4> the system ofT-absorbing subsets
of C. There exists a set Γ e Ή and a function s e L^iT) satisfying
(i) И = 1 on Г and Tf= sT(sf)for anyfe LX{T), and
(ii) if Δ — С\Г, then (I— T)LX(A) is dense in LX(A) in the norm topology.
The partition of С into Г and A is determined uniquely mod μ by (i) and (ii). A
function r has the properties stated for s iff there exists a function le A» CO with
|/| = 1 on Γ, Τ*/ = / on r,andr = sl.

164 Extensions of the Lx-theory
Proof. Γ can be constructed by a simple exhaustion argument: Let ^ be the class
of all sets G e %? for which there exists an A e L^ with Г* A = A on G and Α Φ 0 on
G, and let ν be a finite measure equivalent to μ. Set η = sup {v(G): Gef}.
Let G'n be a sequence in ^ with v(G„) -+η.Γ shall be the (disjoint) union of the
sets Gi = G[, G2 = G'2\GU G3 = G3\(GX и G2\ ... By the construction there
exist functions Awe L^ with T*hn = A„ on G'n and Απ Φ 0 on G^. The function
se Loo CO is defined as 5 = A„/| A„ | on Gn. The desired property of s follows from
lemma 1.9.
If A e Lqo has the property that <(/— T)f A> = 0 holds for all/e Lx (J), then
we have </ Γ* A — A> for all/e Lx (J) and hence Γ* A = A on A. By the definition
of Γ we must have A = 0 on A. Therefore any bounded linear functional
vanishing in (I — T)L1(A) vanishes in LX(A) and (ii) is proved.
If Δ' υ Γ' is a different decomposition of С with the same properties and Be%?
contained in A', then, for any he Lx(B) with Г* А = A, we have </, Г*А — A>
= 0 V/e Lx (5) and hence <(/- Г)/ A> = 0 for all/e Lx (5). As (/- r)Lx (5)
must be dense in Lx (B) by the properties ofzf, we see </ A> = 0 for all/e LX(B),
i.e., A = 0. This proves Α' α Α. Γ' α Γ follows from the maximality of Г, and the
uniqueness is proved.
Now let r g Loo (Г) with И = 1 on Г be given. There exists an / e Loo (Г) with
r = j/ and |/| = 1 on Г. We have Г/= rT(r/) V/e Lx (Г) о 5T(s/) = ЙТ(*//)
V/e Li(r) о T^ = 7T(/g) Vge Li(r). As the last condition is equivalent to
T*/ = / on Г by lemma 1.6., the last assertion in the theorem follows, α
3. Chacon's general ratio theorem. We proceed to apply theorem 1.10 to the
proof of a very general ratio ergodic theorem. A sequence (pk)k^0 of measurable
functions pk: Ω -* R+ is called T-admissible if, for all к Ξ> 0, | Tf\ ^ pk+l holds
for all/e Lx with |/| ^ /?fc. It follows immediately from theorem 1.1 that (pk) is
admissible for a contraction Γ in Lx iff it is admissible in the sense of § 3.3 for the
linear modulus T.
Theorem 1.11 (Chacon). If Τ is an arbitrary contraction in Lufe Lx and (pk)
n — l oo
T-admissible, the ratios Snf/ Σ pt converge to a finite limit a.e. in { Σ ρ{ > 0}.
i=0 i=0
n-1
Proof Let ρ be a strictly positive integrable function. Let us write Sng = Σ T'g.
i = 0
Applying theorem 3.3.7 to Τ we see that it is enough to prove the a.e.-
convergence of Snf/Snp. Let us first look at the case Ω = С For/e LX{T)
the identity Tf=sT(sf) implies Tkf=sTk(sf). We see that Snf/Snp
= sSn(sf)/Snp converges a.e. by the Chacon-Ornstein theorem.
For/e Lx (A) and ε > 0 we may find functions/, ge e Lx (A) with/ = (I— T)fB
+ ge and \\gB\\x <e. The estimate |S„(7- Γ)/| ^ |/| + |Γ"/.Ι ^ |/.| + T"|/.I
and lemma 3.2.3 show that Sn(I- T)fJSnp tends to 0. Using | TgB\ ^ Tf|^e| we

Non positive contractions 165
obtain|lim sup Snf/Snp\ rg limSn\ge\/Snp = d(\gε\μ\(<έ>)/d(pμ\(<έ>).Asε > Owas
arbitrarily small we see that the desired limit exists and is 0 for all/e Ll(A).
To prove the theorem also in the case Ω Φ С first note that, due to | Tlf\
00
rg T'l/I, the sum Σ Tlf converges absolutely a.e. in D. It therefore remains to
i = 0
study the influence of D in С Recall TE = IE+ TIEC. From § 3.3 we know that
00 00
Hc = к Σ (T/D)v and hence also Hc = Ic Σ (TID)V
v=0 v=0
is a contraction in Lv We want to show
(1.4) lim S„f/S„p = lim SnHcf/Snp in С
n-> oo n-> oo
This will complete the proof and provide an identification of the limit. First note
that \Sn(f- Tcf)\ й l/lD| + T"|/1D| implies that
(1.5) lim Sn(f~ T*f)ISnp = 0 on С
n->oo
holds for к = 1, and, by induction, for all k. The identity (3.3.3) remains true for
non positive operators and gives
(1.6) Te = k^ic(TiDy + (TiD)\ (k^i).
v = 0
To estimate the influence of the last summand observe that the identities
Г(Т1в)к = Г1с(Т1в)к + Т1^{Т1в)к+1 = ...
= Г1с{Т10)к + Г-Чс(Т1в)к^ + ... + Т1с(Т1в)к+1-' + (TID)k+i yield
00
(1.7) \IcSn(TIDff\ й S„ Σ /c(T/D)v|/l·
The functions
gk= Σ Ic(TIDYf and hk= Σ /C(T/D)V|/I
v=k v=k
belong to LX(C). Using (1.6) we have
Tkcf-Hcf={TIDff-gk.
By (1.7) we obtain |Sn(T£f- Hcf)\ й Snhk + S„\gk\ on C. Choosing к large the
norm of hk and gk can be made as small as we want, and therefore
lim Sn(hk + \gk\)/S„p = d((hk + \gk\) ■ μ\<€)Ιά{ρμ№
n-> oo
can be made arbitrarily small. Together with (1.5) and the existence of
limn^^ SnHcf/Snp this implies (1.4). α

166 Extensions of the Lj-theory
Notes
The linear modulus of Τ is discussed in the books of Peressini [1967] and Schaefer
[1974]), who treats also the complexification of Banach lattices. Krengel [1963] showed
| Γ| = sup {Re(aT): \ a | = 1}, where sup is taken in the order complete lattice of real
regular operators in Lp. (This was also proved by Schaefer [1974]). Krengel [1963] showed that
the bounded linear operators in L2 need not be majorizable (and form a lattice iff L2 is
finite dimensional). Peressini and Sherbert [1966] noticed that the Hubert transform can
serve as an example in Lp for all 1 < ρ < oo; see also Starr [1971]. An application of | T\ in
Lj is the construction of the ergodic decomposition of T, when | T\ is conservative.
For further related results, see Feyel [1978], and for a semigroup extension of the
notion of linear modulus see § 7.2.
The original proof of Chacon's theorem 1.11 was complicated. Simpler proofs were
given by Akcoglu [1966] and by Akcoglu and Chacon [1970a]. The argument here with
(1.4) seems new. The identification of the limit determined by (1.4), which is the same as
for positive T, seems both simpler and more explicit than that given by Akcoglu and
Brunei.
For a rather short proof of theorem 1.11 in the real case see Gologan [1976].
An alternative proof of Chacon's theorem has been given by Bishop [1968]. He derives
an upcrossing type inequality which is also of independent interest: Let/?0,/71,/72,. ·. and
q0,q1,q2, ...be admissible sequences for a contraction Τ in Lu and let g0, gu ... be a
sequence of integrable functions with
miZftl^l 'Σ gj\, (Pukui).
j=k j=k+l
For each i ^ 0 write ft = gt — qt. For any neN and ωe Ω let Un(co) be the maximum
integer N such that there exist integers
— 1 5Ξ sx(ω) < t^(ω) < s2(co) < t2(co)... < %(ω) < ίΝ(ω) 5Ξ η
with
Σ{Λ(ω): 0ukust}£ ΣίΛΟ») - Α(ω): 0 й к <Ξ /,}, (1 й iu Ю
and
ΣίΛ(ω): Ouk^sju ΣίΛ(ω) - Α (ω): 0 ^ * ^ *,_,}, (2 ^ ι £ Ν).
(A sum over a void set has the value 0. If no such TV exists, then ϋη(ω) = 0). The result of
Bishop is:
Theorem 1.12. The function p0 Un is integrable, and
(1.8) ϊρου^μ^ΠΛ + (-8ο-Ρο)Ί<1μ.
Bishop mentions that the integral on the right side need only be taken over {UN^ 1}, and
that for positive Τ the right side may be replaced by J7J άμ. If, in addition, s1 (ω) is
required to be — 1, then the right side may be replaced by ]f0 άμ. The inequality (1.8) does
not give the identification of the limit.

Vector valued ergodic theorems 167
§ 4.2 Vector valued ergodic theorems
We shall now consider operators Γ acting in the space L\ of Bochner integrable
functions/on (Ω, sf, μ), taking values in a Banach space X with norm | · |. If Γ is
of the form Γ/ = /η for an endomorphism τ of the measure space the pointwise
ergodic theorem for averages Anf shall be deduced from BirkhofFs ergodic
theorem. However, the vector valued generalization of the Dunford-Schwartz
ergodic theorem, due to Chacon [1962], is more subtle.
We begin with a brief description of the Bochner integral. (See also Yosida
[1974], Neveu [1975]). Recall that h: Ω -* X has been called finitely valued if h
assumes only finitely many (distinct) values xl9..., xm and {h = xj has finite
measure for all xt Φ 0. We can then define \1ιάμ — Σχ^μίΑ = хд-
A function /: Ω -* X is called strongly measurable, if there exists a sequence
(Afc), (k — 1, 2,...), of finitely valued functions with \hk —f\ -> 0 μ-a.e., and
Bochner integrable if, in addition, J | hk —f\ άμ tends to 0. We can then define the
Bochner integral \/άμ as the limit of \^άμ.
Passing to h'k (ω) defined as hk (ω) on {| hk | rg 21 /1} and as 0 on the complement
of this set we see that we may always assume |hk\ rg 2|/|, because hk is again
finitely valued, \hk—f\ tends to 0, and \\hk—f\ tends to 0 iff J | hk — f\ tends to 0.
It is then easy to see that a strongly measurable/is Bochner integrable iff J \f\ άμ
is finite. As usual \Β/άμ ·-= J \Β/άμ. It is not hard to check
\\/άμ\^\\ί\άμ.
В В
L* = £ρ(Ω, sf, μ, X) denotes the class of all strongly measurable/, for which the
norm ||/||p := (J \/\ράμ)ι,ρ is finite (1 ^ ρ < oo), and L% is the class of all strongly
measurable/, for which ||/IL*=inf {α: μ({|/| > a}) = 0} is finite.
As in the real and complex valued case these spaces are Banach spaces. (Again,
we do not distinguish functions and equivalence classes of functions).
If μ is finite and s&'0 с si a sub-a-algebra of si, we may define the conditional
expectation of a finitely valued function h = Ydxil{h=Xi} by
E(h|s/0) := T,XiE(l{h=Xi]|si0). For general Bochner integrable/we may define
£'(/|^0)asthelimitinthe|| · W^norraoiΕ{hk\sJ0), assuming\hk\ rg 2|/|again.
Passing to a subsequence of (hk) we can also assume that the limit exists a.e..
Theorem 2.1. Let τ be an endomorphism of a σ-finite measure space (Ω, s4\ μ), Χ a
Banach space, andfe L\. Let Τ be given by Tg = g ° τ. Then Anf converges a.e. in
| · \-norm to a function fe L\ withf=f° τ.
Proof It is enough to prove the theorem in the case (a): μ (Ω) < oo, and in the case
(b), where no non zero finite invariant measure ν <^ μ exists. In case (a) let J
denote the σ-algebra of τ-invariant sets, and Τ the operator g -+ Τ g = g ο τ in
n-l
the real Lv Set An = η'1 Σ (T')v. For any χe X and Be si one has
v = 0

168 Extensions of the Lx -theory
\An{x\B)-E{x\B\J)\ = \x{A'„\B-E{\B\J))\ = \x\\A'n\B-E{\B\J)\.
As hk is finitely valued we see that by the Birkhoff theorem the second term on the
right side of the inequality
tends to 0 for any fixed к when η tends to oo. Consequently, for all k,
limsup|^/-^(/l^)l^limsup|^(/-WI + l^(AJ^)-^(/l^)l·
By the choice of (hk) the last terms tends to 0 for к -* oo. Finally, the triangle
inequality for the norm, and the identity |#°τν| = |#|°τν yield \An(f— hk)\
^ A'n\f— hk\. By the Birkhoff theorem A'n\f— hk\ tends to a limit function with
integral J \f— hk \ άμ. As this can again be made arbitrarily small we have proved
\Anf— E{f\<f)\ -+ 0 a.e. ./= E(f\f) is invariant under τ. In the case (b) one
can replace all conditional expectations in the proof of (a) by 0. Then the identical
argument shows Anf-> 0 a.e.. α
Apparently, the same argument can be applied in more general situations. E.g.
the multiparameter ergodic theorem 6.2.8 yields an extension to Banach valued
functions.
Our next aim is Chacon's vector valued ergodic theorem. We first describe a
filling scheme for £-valued functions. Let ТЫ a linear operator acting in a linear
space L of strongly measurable X-valued functions such that, for all fe L, all
strongly measurable /' with \f'\ <; \f\ belong to L. For any measurable
α: Ω -> U+ set/*" = Min(a, |/|) ·/· \f\~x on {/Φ 0},/e" = 0 on {/= 0}, and
fa+ =f—fa . The filling scheme for/e L and a measurable g: Ω -* R + is defined
inductively by
fo=fg + , do=f'-9 go=g-\d0\,
fn+i = (Tfny»\ dn+1 = (Tfny»-, gn+1=gn-\dn+1\.
Heuristically, gn still measures the size of the hole remaining after step л, the
vector dn is the "matter" lost in the hole exactly at time n. It consumes | dn \ of the
hole #„_! left after step η — 1 .fn is the portion of the matter, which does not drop
into the hole at time η and will be mapped. It is clear that (gn) is a non negative
decreasing sequence and that gn = 0 holds on {\fn\ > 0}. As in the real valued
filling scheme the assertion
(2.1) Pf=f}+ Σ Р-Ык
fc = 0
is trivial for j = 0, and, if it is proved for j9 then
P+1f=Tfj+ i Tu+i)~kdk = (Tfjf^ + (т/у- + Σ Tu+i)~kdk
k=0 k=0

Vector valued ergodic theorems 169
together with the definition of fj+1 and dj+1 completes the inductive proof of
(2.1). Summing from j=0toj=n — l we arrive for η ^ 1 at
n-l n-i j n—1 n-1 n—j—1
(2.2) sj= ΣΛ+ Σ Σ Р~Чк= Σ/;+ Σ τ' Σ dk.
j = 0 j = 0 fc = 0 j = 0 j = 0 fc = 0
η oo
On {gn = 0} wehavedm = Oforallra >Handg = Σ \dt\ = Σ l^jl· Hence, if J?
i = 0 i = 0
00 00
is any subset of { Σ \fn\ > 0}, then g = Σ |^| holds on E.
n = 0 i = 0
00
Lemma 2.2. Le/ Τ be a contraction in L\ and Ε α { Σ |/„| > 0}, ί/геи
n = 0
ί&-Ι4>Ι)^ίΙΛΙΦ·
Proof. The left side is bounded by
00 00
ί ΣΙ4№^ίΣΙ4ΙΦ·
£ i=l i = l
Using 7/„ = dn+1 +/п+1? (и ^ 0), and the contraction property, we find
00
ΙΙ/οΙΙι^ΙΙ^ιΙΙι + ΙΙΛΙΙι^ΙΜιΙΙι + ΙΙ^ΙΙι + ΙΙΛΙΙι^ ··· Ш ΣΙΙ4ΙΙι· π
i = l
If Γ is defined on a space L larger than L\ and the restriction to L\ contracts the
|| · Hi-norm, the lemma still remains true: it is trivial if/0 does not belong to L\. A
contraction Τ in L\ is called an L\ — L%-contraction if also the norm
\\T\\lfO0:=sup{\\Th\\O0:he L^nL^'.WhW^ ^ 1} is bounded by 1. It is now simple
to prove Chacon's maximal ergodic lemma.
Theorem 2.3. Let Τ be an L\ — L%-contraction, α > 0, fe L\, and Ea
= {sup^il^/Ιχχ}. Then
(2.3) \{*-\/«-\)άμ^\\ία+\άμ.
Proof. It remains to show Ea a { Σ I/J > 0} *п the filling scheme with g = a.
n = 0
Now
η — j -1 oo
Ι Σ dk\u Σ l<4l^g = a
fc = 0 fc = 0
and || Г||1>00 ^ 1 yield I^T1 *Σ ^j " Σ dk\uoc for all n.
j=0 k=0

170 Extensions of the Lx-theory
n-l
It therefore follows from (2.2) that Σ I/} I > 0 holds on {\Anf\ > α}, α
j = 0
Now note that an L\ — L% -contraction Γ may be extended in a natural way to
L% (1 Sp< oo): If/is in L* then, for any δ > 0,fd+ is in L\. We define
Tf= lim 7/* + .
<5->o + o
The almost sure existence of the limit follows from \\f3l+ —/*2 + ||oo й 1<5Х — <52|
and || 7Ί|1ί00 rg 1. The functions/a+ belong to Lf. The Riesz-Thorin convexity
theorem (see Thorin [1948]) implies that Γ is a contraction in each L*.
For the proof of a pointwise ergodic theorem X must have some additional
structure. After all, there exists a Banach space X and a contraction T0 in £, for
which the mean ergodic theorem does not hold. Now for a 1-point probability
space Lf consists of all functions χ · H, (x e X). The operator Γ(χ · 1) = (Г0х)И is
an Lf — L%-contraction for which An(xi) need not converge strongly a.e..
Theorem 2.4 (Chacon). Let X be a reflexive Banach space, and Τ an L* — L%-
contraction, then the averages Anf converge for allfe L*, (1 rg ρ < oo), a.e. in the
| · \-norm.
Proof. We need a result of R. S. Phillips [1943], which says that for reflexive X and
ρ > 1 the space Lp(i3, si, μ, Χ) is a reflexive Banach space. (Its dual is
Lq(Q, s/, μ, £*), where/?"1 + q'1 = 1). This together with theorem 2.1.3 shows
that the set of functions h1 + (h2 — Th2) with hl9 h2 e L\ and hx = Thx is dense in
L\. It is then easy to see that, for any η > 0, any/e L* may be written as a sum/
= hx + A'2 - ΓΑ'2 +/r +/", where Ax = Thlf h'2 belongs to L\nL%, and where
/',/" satisfy H/'lli < fMl/ΊΙαο < 4. Set
zl(A) = lim sup \Anh — Amh\.
Clearly Δ (hx) = 0, and A {h2 — ΓΑ2) = 0, because of h2 e L% and cancellation.
Also Δ^")<2η is obvious. The set {J (/') > 2?/} is contained in En
= {sup 14,/'| > η}. Using theorem 2.3 we can see that ημ(Ε'η) is bounded by
||/r \\x < η2. This implies μ(Α (/) > 2η) < η. Combining the estimates with A (/)
й A (h±) + A (H2 - Th'2) + A (/') + A (/") we arrive at μ({Α (/) > 4?/}) < η. As
τ/ > 0 was arbitrary we have proved A (/) = 0, which is equivalent to the assertion
of the theorem, α
Notes
E. Mourier [1953] proved the strong law of large numbers for independent identically
distributed Bochner integrable random variables taking values in a general Banach space.
A. Beck [1963] showed that Kolmogorov's strong law for independent random variables
holds in the ^-valued case iff the Banach space X satisfies a geometric condition, called B-

Vector valued ergodic theorems 171
convexity. This led to a lot of further work connecting probability theory and the
geometry of Banach spaces, see Padgett and Taylor [1973], Woyczynski [1978]. Beck and
Schwartz [1957] proved a vector valued random ergodic theorem for reflexive X, now
contained in theorem 2.4. Theorem 2.1 and several generalizations appear in the work of
Tempel'man [1972], [1967]. The simple reduction to BirkhofFs theorem has been used in
the paper of Landers and Rogge [1978] to prove a generalization where/is allowed to
assume values in a Frechet space.
Call a sequence (Fn)n^ x of Bochner integrable functions on a probability space (Ω, л/, μ)
with values in a Banach lattice X a subadditive vector valued process (for the endomorph-
ism τ) provided for all natural k, η we have Fn+k 5Ξ Fn + Fk ° τ". X is called countably order
complete if sup„^ 1 xn exists for any sequence (xn) bounded above by some yeX.Xis said
to have order continuous norm if X is countably order complete and every decreasing
sequence zneX with zn ^ 0 is norm convergent.
Theorem 2.5 (Ghoussoub-Steele [1980]). For countably order complete Banach lattices X
the following are equivalent:
(i) For every X-valued subadditive process Fn ^ 0 we have norm convergence ofFn(co)/n
for a.e. ω;
(ii) X has order continuous norm.
For an alternative proof and for related results see Davis-Ghoussoub-Lindenstrauss
[1981], and Derriennic-Krengel [1981].
The technique of the vector valued filling scheme has been extended to a more general
situation, permitting a unified proof of Chacon's maximal theorem 2.3 and the
(decreasing) martingale theorem by A. and С Ionescu Tulcea [1963]. One of their results is:
Theorem 2.6. Let Xbea Banach space. Let I = T0, Tu ..., Trbe L* — L%-contractions with
Tj+l Tj = Tj+1, (j — 0,..., r — 1), α > 0, andfe L*, (1 ^ ρ < oo). If F is a measurable set
containing {\f\ > a} and contained in
{sup{\(T}o + ... + T}n-i)f/n\:0^j<r,n^l}>a}
then
a/i(/0^jl/№<oo.
F
Hasegawa, Sato and Tsurumi [1978] derive, among other results, a continuous parameter
extension of theorem 2.4 and replace the assumption || T\\loo ^ 1 by sup {|| Гп||1>00} < oo.
Related papers are those of Chersi and Invernizzi [1976] and Kopp [1975].
Flytzanis [1976] has given the identification of the limit in the vector valued random
ergodic theorem of Beck and Schwartz.
Pop-Stojanovic [1972] has proved a vector valued ergodic theorem for weakly
integrable strictly stationary sequences.
Jajte [1968] showed: Let J7 be a unitary operator in a Hubert space Ж, Xan Ж -valued
random variable with square integrable norm. Put Χ}{ω) = Uj(X(co)). If Y0, Fl3... is a
sequence of independent Ж -valued random variables, with Yj distributed like Xj, then
n'1 (Y0 + ... + Υη-ι) (ω) converges in norm for a.e. ω. This was generalized and simplified
by Al-Hussaini [1974].
Yoshimoto [1979] has vector valued estimates of the L \ogmL type; see §1.6.

172 Extensions of the Lx-theory
§ 4.3 Power bounded operators and harmonic functions
We return to the study of positive linear operators Tin the real Lv It is a simple
exercise to show that such operators are bounded but we do not assume now that
Γ is a contraction.
By the usual monotone extension T/and Г*/аге well defined (possibly infinite
valued) for all non negative measurable/; see § 1.6.
If e is superharmonic, E={e> 0}, and/e L\ {Ec\ then Je(Tf) = l(T*e)f
<, jV= 0 shows Tfe L\ (Ec). In other words: Ec is Γ-absorbing. For any/^ 0
we have \{Τ/)ά{βμ) = \{Τ/)βάμ = 1/(Τ*έ)άμ й Ι/βάμ = \ίά(βμ). Hence Γ is
a positive contraction in Lx (£", βμ) = Lx (βμ), and we can apply the results
previously derived for positive contractions to Γ, although Tneed not be a contraction
in Lx (μ). E.g., we can infer that Snf/Sng converges (e//)-a.e. on {S^g > 0}, and
hence μ-a.e. on En {S^g > 0} for any/e Lx (βμ), ge L\ (βμ). Here we have not
assumed ее L^. But if г is in L^, any μ-integrable/is (e//)-integrable, and the
Chacon-Ornstein theorem holds on Ε in the usual formulation. This may have
motivated the search for harmonic functions e ^ 0 for which Ε is as large as
possible.
By an exhaustion argument we can find a decomposition of Ω into two disjoint
measurable sets Υ and Z, such that the support {e' > 0} of every harmonic
e' e L+ is contained in Υ and there exists a harmonic e e L + which is strictly
positive on Y. Obviously the decomposition is determined uniquely mod nullsets.
It is called the Sucheston decomposition of Ω. For reasons which will be plain from
theorem 3.1 below У is called the remaining part of Ω, and Ζ the disappearing part.
The full statement of the following theorem was first proved in the work of
Derriennic and Lin [1973]:
Theorem 3.1. Let Τ be a Cesar о bounded positive linear operator in the real Lv
Then
N-l
(i) feL^Z) implies lim Ν'1 Σ ll^n/lli=0;
N-►00 n = 0
(ii) liminf \\Τη/\\χ >0 holds for all non-zero f e L\{Y)\
(Hi) the harmonic e with Y= {e > 0} can be chosen in such a way that \h\ is
bounded by a multiple of e for each harmonic he L^;
(iv) for all he L^, A%h converges to 0 a.e. in Z;
(v) Tnf/n converges in Lx-norm to 0 for all fe Lx.
If Τ is even power bounded the assertions (i) and (iv) may be replaced by the stronger
assertions (i'): || Tnf\\x -* Ofor allf allf в Lx (Z), and(W)\ ton, T*nh = 0 a.e. in Ζ
for all he L^.
Proof If Μ is a bound for all norms 11 An \ | the non negative functions gN --= A% H
are bounded by M, and g = lim sup gN is in L^. Writing g = limk (supN^k gN) we

Power bounded operators 173
N
see T*g ^ lim sup T*gN = lim sup TV"1 Σ Τ*ηί = g. Let e be the limit of the
k = l
increasing sequence T*kg. As g is bounded by Μ and e is also the limit of the
sequence A%g, we obtain ЦеЦ^ й Μ2. It is clear that e is harmonic. Set Υ
= {e > 0}. When we show (iii), Υ must be the remaining part of Ω as defined
above. We may assume \h\ rg 1. Now T*h = h implies T*h+ ^ h+ and Γ*/Γ
^ A- and hence also T*\h\ ^ \h\. (iii) therefore follows from \h\ rg lim Г*к|А|
= lim^S|A| ^g^e.
(ii): For any /е£^(У) with /Ф0, we have 0 < \/βάμ = \/Τ*ηβάμ
= <Tnf,e}u\\Tnf\\1-\\e\\O0.
(i): For any fe Lx (Z),
lim sup TV"1 V II Tnf\\i й lim sup <AN\f\J}
N-►00 n = 0
= lim sup <|/|, gN> ^ <|/|f lim supg]V> ^ <|/|, g> g <|/|, e> = 0.
(iv): We may assume \h\ rg 1. The definition of г yields
0 rg J lim supyl* |A |άμ <; J lim supg„i^u = J gdμ rg J βάμ = 0.
ζ ζ ζ ζ
(ν): We may assume/^ 0. Set an = || Τη/\\χ. Then, for A: ^ л, we have
^/(fc + i^i/t + i)-1 ς α,ΗΚί + ΐΓ'ΣΓ^-'ΛΐΙι
i=n—k i=0
Summing over A: we obtain
(я»*Σ (А^Г^Мя"1^ «„-fe^M^^oo.
fc = 0 fc=0
n-1
As Σ (^ + I)1 tends to infinity the assertion ajn -* 0 follows.
fc = 0
If Τ is even power bounded e is defined by
e = limr*k(limsupr*wH)
к п
and the same proof shows that (i') and (vi') hold, α
When Τ is even a positive contraction in Lx the restriction of Τ to LX(C) is
Markovian and (i) implies that Ζ is contained in the dissipative part D. If τ is an
endomorphism of (Ω, sf9 μ), and Γ is given by Tf = f° τ, we clearly have Ω = Υ
and see that D = Υ is possible.
Sucheston [1967b] has given an example of a power bounded Γ with Ω = Ζ
such that the ratio's Snh/Sng need not converge a.e. on {S^g > 0} for h,geL\.
This may be seen also in the following simple example, which Fong [1970] gave
in order to show that even the Hopf decomposition may fail on Z: Take Ω

174 Extensions of the Lx -theory
= {0,1, 2,...} and let μ be the counting measure. Define ГЬу (Tf) (0) = Σ ДО,
i = l
and (Tf) (J) =f(J+ 1) for/ ^ 1. For all л £ 1, (Г-/) (0) = £ ДО and (Г"Л О)
i = n
00
= /C/ + fl)fory ^ 1. It follows that || TYHi is bounded by 2 Σ 1/(01 ^ 211/Hi.
Hence || Τη\\χ S 2, (n 11), and lim || Tnf\\1 = 0 yields Ω = Ζ. It is simple to
check that (Saog)(0) = ao holds for g defined by g(0 = (/+1)"2, and
(iSc»/) (0) < oo for any/with finite support. If i1 < i2 < /3 < ... is increasing
sufficiently fast and h is defined by h{i) = 0 for i in [iv, /v+1 [ with ν even, and h{i)
— g(i) for / in [/v, /v+1[ with ν odd, the ratios Snh/Sng diverge in the point 0.
A. Ionescu Tulcea and M. Moretz [1969] proved that, for power bounded Τ
and ge L\, there is no subset A of Ζ η {S^g = 00} with μ (A) > 0, such that
Snf/Sng converges a.e. on A for all/e Lx.
If Γ is Cesaro bounded and/e Lx (μ), ^„/converges stochastically in У
(Exercise). Fong [1979] proved that n~l Fn converges stochastically in У for superaddi-
ti ve (Fn) with sup ||w_1/5illi<oo, and gave an example of a power bounded Τ and
an/e LX(Y) for which yl„/does not converge stochastically in Z.
We shall now use the Sucheston decomposition to prove:
Theorem 3.2. Let Τ be a Cesaro bounded positive linear operator inLl9 and ρ e Lx
strictly positive. The following conditions are equivalent:
(a) for every h e L^, A* h converges a.e. in Ω;
(b) for every fe Ll9 Anf converges in Lx;
(c) for every Ae si, the sequence </?, A* lA} converges.
Proof. Passing to the new reference measure μ' = ρ μ we may assume μ(Ω) < oo
and ρ = 1. (a) => (c) is trivial, (c) => (b): (c) means that Ani converges weakly in
Lx. By (v) of the previous theorem ||л_1ГиИ \\x -* 0. Now the mean ergodic
theorem 2.1.1 shows that Ani converges strongly. Therefore the sequence (AnV)
is uniformly integrable. By approximation we find that for all/e Lx the sequence
(Anf) is uniformly integrable and hence conditionally weakly compact. Theorem
2.1.1 implies (b).
(b) => (a): Construct g, e and Fas in the proof of theorem 3.1. Let и be the strong
limit of (An\). Clearly 0 ^u = Tu. Because of
\(T*h)udμ = \Η(Τυ)άμ = \hudμ
Γ* is a positive contraction in L1(u· μ). As L^ is contained in Σχ^μ), the
Chacon-Ornstein theorem implies the convergence (u/j)-a.e. of
A*h = e · (S*h/ne) = e(S*h/S*e) on {e > 0}, or, equivalently, the convergence
μ-a.e. on Υ η {и > 0}. In analogy to the definition of g and e we now define

Power bounded operators 175
gh = lim sup^4*A, gh = lim infA%h,
eh = lim A%g\ eh = lim A%gh.
Then eh and eh are harmonic and T*gh ^ g\ T*gh ^ #й yields eh^gh^gh^ eh.
As ей — eh is harmonic and therefore bounded by a multiple of г we obtain gh = #л
on Ζ = {г = 0}. Together with the result above we have gh = gh on {u > 0}.
Observe that <w,r*kl{M=0}> = <7*и,1{« = о>> = <w,l{M = 0}> = 0 yields that A* ν is
0 on {u > 0} for all ve L^ ({u = 0}). We can infer that eh = eh holds on {u > 0}.
Consequently we have <H, eh - eh} = <1, A%(eh - eh)} = <^B1, eh - eh}
= <w, eh — eh} = 0. We have proved гй = eh μ-a.e., i.e., (a) holds, α
In the case of contractions theorem 3.2 follows from results of Helmberg [1972]
and of Lin and Sine [1977]. For point maps, see Dowker [1947] and Wright
[1960]. The general case is due to Sato [1978a]. It seems useful to state
Helmberg's criterion separately:
Theorem 3.3. If Τ is a positive contraction in Lx the condition (a) in the previous
theorem is equivalent to
(d) C\C is empty and T*klD tends to 0.
Proof. It remains to prove the much simpler equivalence of (b) and (d). Assume
(b). From the remarks on Lx-convergence in § 3.3 we know that (b) implies C\C
= 0. If the limit of the decreasing sequence T*k 1D is not 0 there exists a non zero
fe L\ (D) for which (Anf 1D> = </, A* 1D> does not tend to 0. The Γ-invariant
limit /of A „/has the property </ 1D> > 0. This contradicts CcC.On the other
hand(d) implies (b) because of uniform integrability and a.e.-convergence of the
sequence Λ/οη С together with (Anf 1D> = </ A* 1D> -* 0, (fe L\). и
Notes
For power bounded T, Sucheston [1967a, b] proved (i') and (ii), and A. Ionescu Tulcea
and M. Moretz [1969] proved (iv') in theorem 3.1. R. Sato [1973] derived the
decomposition Υ + Ζ in the Cesaro-bounded case independently of Derriennic-Lin. Sucheston
actually used the condition <^sup < Tnf h} < oo V/e L\ > for fixed he L* and constructed
partitions Ω = Yh + Zft. For h = H his condition reduces to power boundedness and the
partition agrees with Ω = Υ Λ-Ζ. Fong [1970] showed that his approach works also in Lp,
(1 ^ ρ < oo), and studied the existence of Γ-invariant functions. For this problem see also
Derriennic-Lin [1973], Ornstein [1970], Fong and Lin [1976]. Sato [1977] studies the
existence of invariant functions for ergodic semigroups in the sense of Eberlein and gives
many references. Continuous parameter results appear in Fong and Sucheston [1971],
Sato [1977a], and Fong [1979].
Using theorem 3.2, Sato [1978a] proved:
Theorem 3.4. Assume μ(Ω) < oo. Let Τ be a bounded linear operator in L1 such that the
linear modulus Τ is Cesar о bounded both in Lx and in L^. Then Anf converges a.e.for all

176 Chapter 4: Extensions of the Lx-theory
Derriennic and Lin already proved this for Τ ;> 0, 7Ί = 1 (and gave the identification of
the limit) and they showed by example that v4„/need not converge a.e. for all/e Lx; see
also Kubokawa [1972a], Dunford [1980].
Generalizing a result of Ornstein [1969], Fong and Sucheston [1973] have given an
extension of the Jamison-Orey theorem to power bounded operators: If Γ is a power
bounded positive operator in L1 with Ω = Y= {e > 0}, and/e L\ implies {5^/= oo}
= {SODf> 0}, then lim || T'g^ = 0 holds for all ge Lx with lim |k(Tn#)lli = 0.
Derriennic and Lin have given a simpler proof.

Chapter 5: Operators in C(K) and in Lp,
The main results in our treatment of Markov operators in C(K) are the theorem
of Jamison deducing uniform convergence of Snf from pointwise convergence
for irreducible S, and the strong law of Breiman for Markov processes with
arbitrary initial distributions.
Concerning Lp, we present Akcoglu's pointwise ergodic theorem for positive
contractions and the dominated estimates of Stein for self-adjoint operators.
§5.1 Markov operators in C(K)
1. Mean ergodicity and unique ergodicity. Let КЫ a compact Hausdorff space. A
linear operator S in the Banach space X = C(K) of real or complex continuous
functions on AT with ||/|| = sup {|/(ω) |: ω e K) will be called Markov operator if
it is positive and satisfies SI = 1. E.g., a continuous map τ: К -> К induces a
Markov operator by 5/(ω) =/(τω).
We give only a brief introduction to the convergence theorems for Markov
operators. For the basic notions in point set topology and functional analysis we
refer to Dunford-Schwartz [1958] and Yosida [1974].
By the Riesz representation theorem each continuous linear functional ν on X
is of the form </, v> = J/rfv, where ν is a signed measure on the Baire σ-algebra
3F in K. Thus X* is the space Ji of signed measures on (K, IF). Ji£ denotes the
set of probability measures on #\
A Markov operator S defines a stochastic kernel by Ρ(ω, ·) = S* εω9 where εω
is the point mass 1 in ω. We have
(1.1) 5/(ω) = <S/, βω> = </, Ρ(ω, ■)> = f/(ι?) Ρ{ω, άη).
The continuity of Sf implies that the map ω -* Ρ {ω, ·) of К into Jl? is
continuous in the w*-topology of X*. Conversely, if an arbitrary stochastic kernel
{Ρ(ω, Α): ω g К, А в ^} is given, and we define Sf by (1.1), then the w*-
continuity of the map ω -* Ρ (ω, ·) is sufficient for Sfe C(K). [We remark that,
by (1.1), S has the interpretation of the operator T* in section 3.1].
We use the notation Anf= An(S)f= n~x{f+ Sf+ ... + S"1-1/) here.

178 Operators in C{K) and in Lp
Proposition 1.1. Let S be a Markov operator in X = C(K) and fe X. Then the
following assertions are equivalent:
(i) Anf converges strongly,
(ii) Anf converges pointwise and the limit f is continuous;
(Hi) The functions Anf are equi-continuous.
Proof The dominated convergence theorem implies that a bounded sequence/,,
in X converges pointwise to a continuous limit function /iff it converges weakly
to / Thus the equivalence of (i), (ii) and (iii) follows from the mean ergodic
theorem, and the Arzela-Ascoli theorem, α
Any Markov operator S admits at least one invariant probability measure: To see
this take any Banach limit L and any ω e К and put μ(/) = L(((Snf εω»£°=ο)·
The properties of Banach limits imply μ(Ί) = 1,μ^0 and μ (Sf) = μ(/). Thus
the linear functional μ belongs to M?', and it satisfies S* μ = μ.
S is called uniquely ergodic if S admits only one invariant probability measure,
and mean ergodic if ^„/converges strongly for all/
Proposition 1.2. Any uniquely ergodic S is mean ergodic.
Proof Let ν Φ 0 be a fixed point of S*. v+ satisfies S*v + Ξ> S*v = ν and S*v +
^0. Hence S*v+^v\ Now <1, v+> = <51, v+> = <1,5*v + > yields S*v+
= v+ and therefore also 5*v" = v" It follows that v+ and v" are non negative
multiples of the unique invariant probability measure μ. ν φ 0 then implies
<v, H > Φ 0. Thus the fixed points of S separate the fixed points of S*. Now apply
theorem 2.1.4. α
Proposition 1.3. A mean ergodic S is uniquely ergodic iff the limit f of Anf is
constant for each f
Proof If/is constant for all/and v1? v2 are S^-invariant probabilities, then
</ vf> = </ A* vf> = <Anf v£> = </ vf>.
Thus </ νχ> = </ v2> implies </ vx> = </,_v2>. As/was arbitrary vx = v2.
Conversely, if an/and two points ωΐ9 ω2 with/(ω1) Φ/(ω2) exist, the sequences
Α%εω. converge weak * to S^-invariant probabilities vt with
</ vt} = lim </ A% εωι> = </ εωί> = /(ω,). α
A continuous τ : К -> К is called uniquely ergodic if the operator Sf = f° τ is
uniquely ergodic. By theorem 1.2.7 the irrational rotations of the torus are
examples. Many other examples can be found in the books on topological
dynamics.

Markov operators in C(K) 179
For general Markov operators S in C{K) even pointwise convergence of Anf
may fail. E.g. let AT be the unilateral product space {0,1} χ {0,1} χ {0,1} χ ...
and τ the shift in K. If g is the indicator function of the set of co's with first
coordinate 0 and η an element of AT in which the relative frequency of zero's does
not converge, then Angty) does not converge.
A point ω is called quasi-regular for S if Anf(co) converges for all/e X. If all
points are quasi-regular and the limit function /of Anf is continuous, then S is
mean ergodic by proposition 1.1. However, there exist Markov operators for
which all points are quasiregular but / is discontinuous for some /. [Take
K= [0,1] and τω = ω2].
Induced contractions in Lp. Let μ be an S^-invariant probability measure on
(K,^) and Lp = Lp(tf,^,/i). Then <|/|,/i> is the Lrnorm \\f\\x offeX.
It follows from <|/|, μ} = <|/|, S*μ} = <5|/|, μ} ^ <|S/|, μ} that \\Sf\\x
^ ||/||1β 51 = 1 implies HS/]^ ^ H/Ц^. Thus S is a positive Lx -
L^-contraction on X a Lv As X is dense in Lx by the Daniell-Stone theorem (see e.g. Bauer
[1981]), the operator S may be extended by continuity to Lx and the extension is
an Li — Lqq-contraction ^i (depending on μ). Let Sp be the restriction of Sx to
Lp cz Lx. By lemma 1.7.4 Sp is a contraction in Lp.
Remark 1.4. The ergodic theorem of Hopf (1.7.3) now implies that
An(Sx^converges μ-a.e. for all/e Lv In particular, ^„/converges μ-a.e. for all/e X. If AT is
compact metric, £ is separable. As the convergence of Anf (ω) for all/in a dense
subset of X implies the convergence of Anf(co) for all/e £ the set of quasi-regular
points has full μ-measure for each invariant μ e Μ\.
2. Irreducible Markov operators. A Markov operator S in X = C(K) is called
irreducible if for each non negative somewhere positive/e 3E and for each сое К
there is an n with Snf((o) > 0.
It may help to see an equivalent probabilistic condition: Let Pn (·, ·) denote the
kernel belonging to Sn and describing the «-step transition probabilities. S is
irreducible iff for all non empty open U and all ω e K, there exists an n with
Ρη(ω, U) > 0. [If the probabilistic condition is satisfied and/, ω are given, take £/
= {/> 0}. Then 5"/(ω) = ]У(т/) i>n(ω, </ι/) is positive. IfS'is irreducible and ω, С/
are given find (by the normality of AT) a continuous/ ^ 0 vanishing in C/c and = 1
in some point of U. Next find n with Snf(cj) > 0. Then Ρ"(ω, £/) is positive.]
If iSis irreducible, only the constant functions are iS-invariant. Otherwise there
would exist a non negative non constant/= 5/which is 0 in some point ω. But
then Snf((D) > 0 would contradict /= Snf. The transformation ν.ω-^ω2 in
[0,1] is an example showing that the converse does not hold. It is not irreducible,
but its only iS-invariant functions are the constants.
By proposition 1.3 a mean ergodic S is uniquely ergodic iff Sf — /implies
/constant. Thus a mean ergodic irreducible operator is uniquely ergodic.

180 Operators in C{K) and in Lp
[If τ: Κ -* К is a homeomorphism, then S with Sf = f°T is irreducible iff τ is
minimal in the sense of topological dynamics, i.e., iff there exists no non empty
proper closed subset А с К with τΑ = A. There exist minimal transformations
which are not uniquely ergodic and therefore not mean ergodic, and there exist
uniquely ergodic transformations which are not minimal; see e.g. Oxtoby [1952],
Keane [1968], Grillenberger [1976].]
Theorem 1.5 (Jamison [1970]). If К is a compact Hausdorjf space, S an
irreducible Markov operator in X = C(K), andfe X satisfies
(1.1) lim 5"/(ω) = 0 for all сое К
η
then ||57Ί|-*0.
Proof We first assume AT compact metric with a metric d. For a bounded sequence
(A) in X put
««(ω) = lim supw (sup {|/η(η) |: d(a>9 η) = l//w}),
and letw(co) be the limit of the decreasing sequence wm(co). For ε>0
and large m we have и (ω) + ε = ит{оэ). If (cok) is a sequence in Attending to ω and
/c is so large that ά{ω, cok) < \βπι holds, then wm(co) = u2m{(ok) = w(cok). Hence
lim supku(cc>k) = w(co) follows and и is upper semicontinuous. Thus и assumes its
supremum a. We now prove
(1.2) ||(|/Jva)-a||-*0, (n-oo).
If this fails there exist sequences ик -► oo and cok, and a number <5 > 0 such that
fnk(coh) = α + <5. We may assume that cok converges to some ω by taking
subsequences. Fix any m. For к large d(cok, ω) < 1/ra, and hence sup {\f„kQi) |: ί/(ω, ?/)
^ 1/ra} = α + <5. Thus wm(co) = α + δ. As га was arbitrary, w(co) = α + <5, a
contradiction.
We shall need
Lemma 1.6. If /„(ω) -> 0 /or еясА ω, /Леи {ω: и(оэ) > 0} w a set of the first
category.
Proof. For each j =1,2,... the sets
Fj={a>sK:u((o)^llJ}
are closed since и is upper semicontinuous. Fix j and suppose that F} has non
empty interior G. Then there is a closed set Fc G with non empty interior. The
sets
Hi = {coeF: \fk(co)\ = 1/2/ for all jfc = /}, (/ = 1,2,...)
are closed and their union is F. By the Baire category theorem applied to the

Markov operators in C(K) 181
complete metric space Fat least one of the sets Ht must have a non empty interior
G'. As each \fk\ is bounded by 1/2/ for к ^ i on #i? we have w(co) ^ 1/2/ on G'.
This contradicts to С с /J.
It follows that each Fj has empty interior. As the set {ω: w(co) > 0} is the union
of the sets Fj9 it is of the first category, α
We now put/„ = Snf If α is 0 the assertion of the theorem follows by (1.2).
Suppose α > 0. As и is upper semicontinuous the set Ε = {ω: u{d) = α} is non
empty and closed.
Lemma 1.7. Ε is absorbing, i.e., Ρ (ω, Ec) — 0 holds for ω e E.
Proof. Assume there exists an ω e Ε with Ρ (ω, Ec) > 0. Then there exists a closed
set F' cz Ec with Ρ (ω, F') > 0. Hence there exist two disjoint open sets G, G' with
£cGandFcG'.
It follows from и (ω) = α that there are sequences cofc -* ω and лк ->oo
with \Snk+1f(cok)\ -* a. The w*-convergence of Ρ(ωλ?·) to Ρ (ω, ·) yields
limfcP(cofc, С?') ^Ρ(ω, С?') > 0. Let Ρ denote the closure of G\ and у
= Ρ(ω, G/)/2. Then Ρ(ωλ, Ρ) ^ у > 0 holds for all large enough k.
Put β = sup {w0/): ijei1}. Since Ρ с Pc, and the supremum is assumed by the
upper semicontinuity of u, we have β < a. Let 2ε = α — β. We have
(1.3) 5""+1/(ω,) = J S*mP(<ok, άη) + J 5"*/(|0Р(а>к, άη).
F Fc
Observe that P(cok?Pc)rgl— у holds for large к and that (1.2) implies
|S"/(!0|va-*a.
Now assume that | Snkfty) | ν (α — ε) tends to α — ε uniformly in P. Then (1.3)
leads to the contradiction
α = lim|5'nfc+1/(cofc)| <; y(a — ε) + (1 — у)α = α — γε<α.
Thus we can find a subsequence (rj) of the sequence (nk) and a sequence τ/7· e F
with | S""J tyj) | > α — ε. We may assume that the sequence η} converges. The limit η
necessarily belongs to P. But then ιι{η) ^ lim sup | Srjfyj) | ^ α — ε > α — 2s = β
leads to a contradiction to the definition of β. α
Now the proof of theorem 1.5 for compact metric AT follows easily: As Ρ (ω, Ρ)
= 1 holds for all ω e P, an inductive argument shows Ρ"(ω, Ρ) = 1 for all ω e Ε
and all n. By the irreducibility of S the non empty, closed set Ρ must therefore
coincide with K. Hence и{оэ) = α > 0, and this contradicts to lemma 1.6.
The general case. It remains to eliminate the additional assumption that К is a
metric space.
Put At = {1,/, TfT2f...}, and let,42 be the set of all finite products

182 Operators in C{K) and in Lp
Si' Si · · · Sk w^h Sie ^1· If ^3 denotes the set of all rational linear combinations
of elements of A2 and A the norm closure of A3, then A is a separable closed
subalgebra of £ with ТА с A.
Let £be the space of equivalence classes of elements of AT with ω ~ ω' iff #(ω)
= g(co') holds for all g e Α. Κ is a compact Hausdorff space with the quotient
topology and to each/e A there corresponds an/e C(K) in an obvious way.
Let A be the set of all/with/e ^4. Then A is a closed subalgebra of С (К)
containing A and separating points. So A = С (AT) by Stone-Weierstrass. As Л is
separable, Λ* is metric. Since 5^1 <= A, the restriction of S to yl lifts to a Markov
operator 5 on C(X) for which Snf is the image of S^/under the isomorphism. It
remains to check that the assumptions for S yield the analogous assumptions for
S. So || £"/11 -* 0 by the metric case, and thus \\Snf\\ -* 0. α
As an application we show a strengthening of the Jacobs-Deleeuw-Glicksberg
decomposition of X for the present special case.
Theorem 1.8. Let К be a compact Hausdorff space and S a weakly almost periodic
irreducible Markov operator in X. If f is a flight vector, then ||5n/ll -* 0·
Proof. There exists a sequence щ-^оо with Snif -* 0 weakly and hence pointwise.
As S is weakly almost periodic, S is mean ergodic, and there exists a unique S*-
invariant probability μ. Now \\8η^\άμ -* 0, and the fact that S contracts the
Lx (μ)-ηοπη implies J | Snf\ άμ -* 0.
Assume S^/does not converge to 0 weakly. Then there exists 0 φ g e X and a
sequence m{ -* 00 with Smif -* g (weakly). Then \\8γη^\άμ ->\\£\<1μ. Thus a
contradiction results if we show \\g\dp > 0. By the irreducibility of S, Ω is the
union of the sets {Sn\g\ > 0}. Hence (Sn\g\9 μ} > 0 holds for at least one n. But
the S^-invariance of μ gives (Sn\g\9μУ = <|g|, μ>. Thus 5"1/ converges to 0
weakly, and the assertion follows from theorem 1.5. α
Corollary 1.9. Any weakly almost periodic irreducible Markov operator S in X
= C(K) is strongly almost periodic.
Proof X = Xrev © Xfl; see section 2.4. α
3. Breiman's strong law. Now we turn to the proof of Breiman's strong law of
large numbers for processes of the form/(X0), f(Xi)9... where/is continuous
on the state space К and X09Xi9... a Markov process with arbitrary initial
distribution. We shall use standard probabilistic concepts freely here.
Let (Kn9 #„) = (К, У\ (« = 0,1,2,...), Ω = Κ0χΚχχ..., and for ω
= (ω0, ωΐ9...) g Ω let Χη{ω) = ωη be its л-th coordinate. If μ is any probability
measure on (K0, #Ό) there is a unique probability Ρ on the product σ-algebra
si = #0 ® ^1 ® · · · f°r which the process X09 Xl9... is Markov with transition

Markov operators in C(K) 183
probabilities P(Xn+1 e Bn+1 \Xn = ωη) = Ρ(ωη, 2?w+1)and initial distribution
P(X0 e Bo) = μ (Bo)- The following result is essentially due to Breiman [I960]:
Theorem 1.10. Let Kbe a compact Hausdorjf space, and S a mean ergodic Markov
_ n-l
operator in X = C(K). Forfe X letf= lim Anf. Thenn~l Σ f(Xk(co)) converges
fc = 0 _
P-a.e., and the distribution of the limit is the same as the distribution μί of/under μ,
where μ is the w*-limit of the sequence Α* μ. (Recall Ρ(ωη, ·) := Ξ*εωη).
The proof will be split into some lemmas:
Lemma 1.11. Ifh = SheX, then h (Xn) converges P-a.e. to a limit Υ distributed as
/v
Proof As Ρ(ωΒ_ΐ5 Bn) = P(Xn e Bn\Xn^ = ω^^ holds for all η, ω„_1? and Bn,
we have E(g(Xn)\Χη-γ) = (Sg) (Χη-Χ) for all geX. Using the Markov property
we find
E(h(Xn)\X0,..., *■„_!) = E(h(Xn)\Xn^) = (Sh) (Xn^) = h(X„^) a.e..
Thus h(X„) is a bounded martingale and converges a.e. to a limit Γ. (See e.g.,
Loeve [1960: 393]). For any bounded ge C(X), g(h(Xn)) -* g( Y) a.e.. Hence
n~l "Σ E(g(h(Xk))) - E(g(Y)). But
fc = 0
«_1 "Σ E(g(h(Xk))) = αΓ1 "Σ j£{g°h(Xk)\X0 = ω0}μ(ί/ωο)
fc = 0 fc = 0
fc = 0
This implies £"(#( У)) = <g ° А, μ> for all bounded g e C(X), which is equivalent
to the assertion that Υ is distributed under Ρ as h = /Tis distributed under μ. α
Lemma 1.12. Le/ ω0? ω1?... be a sequence in K. Assume
(1.4) и"1 "Σ {Sg(o)fc) - g(o)fc+1)} ^Q forge {f, Sf, S2f,...}.
k = 0
Then
(1.5) /Ю-и"1 "iVfaJ-O, (и-оо).
fc = 0
Proo/. Using "Σ {^+Υ(ω,)-/(ω,+1)} = "ς {^'+1Д<%> - S'/fo+1)}
fc = 0 fc = 0
n-l
+ Σ {Sjf((ok) —/(Wk+i)} + Sjf(Mr) — $]/(ωο) an inductive argument yields
fc = 0

184 Operators in C{K) and in Lp
n-l
η'1 Σ {S'fdcoJ -f(cok+1)} -* 0 for; ^ 1. For; = 0 this is trivial. Hence
fc = 0
«_1 "Σ {Amf(«>u) -f(<ok+i)) - 0
fc = 0
holds for each m ^ 1. Since ylm/converges uniformly to/we obtain (1.5). α
We can now derive the theorem from the following strong law for martingale
differences: If s$'0 с ^ с... is an increasing sequence of σ-algebras in si and
Z0? Z1?... an L2-bounded sequence of random variables for which Z„ is sin-
measurable, then
η-1 *Σ {Zk+1-E(Zk+1\stk)}^0 a.e.;
fc = 0
see e.g. Loeve [1960: 387]. Let sik be the σ-algebra generated by X0,..., Xk and
Zk = g(Xk). Then E(Zk+1\sik) = S£(Xk) shows that (1.4) is satisfied for P-
almost all sequences ω = (ω0? ω1? ω2?...)· Hence
(1.6) /Щ-л-'Х/га^О, (и-оо)
fc = 0
а.е. It remains to apply lemma 1.11 with h = f α
Remark 1.13. (1.6) shows that almost surely the sequences f{Xn) and
n-l
n -1 Σ /№) have the same limit.
fc = 0
We close this section with a result of Choquet-Foias [1975] in which S need not
be Markovian or a contraction.
Theorem 1.14. Let К be a compact Hausdorjfspace, and let S be a positive linear
operator on X = C(K). If Sni converges to 0pointwise, then \\Sni || -> 0.
Proof. By the compactness of К there exists an m and δ > 0 with
inf {5*1 (ω): 1 й к й т} й 1 - 2δ for each ω β Κ. If θ > 0 is sufficiently small,
the operator i? = S' + 0/ satisfies t;:=inf{i?k1l: 1 <; A: <; m} < 1 — (5. Now ν
йт{{ЯН:0йкйт-1} implies 7?г; ^ г;, and hence Sv й (1 - θ)ν. By
induction Snv й (1 - 0)Bt>. On the other hand t; ^> 0m1. Together we obtain
Sni й e~mSnv й в~т{\ - 0)WH and this tends uniformly to 0 for η -* οο. α
As abstract M-spaces with unit (like L^) have an isomorphic representation as
spaces of continuous functions on a compact space, the results of this section may
also be applied to them. Of course pointwise convergence to a continuous limit
must be translated into weak convergence then.

Markov operators in C(K) 185
Notes
Krylov and Bogolioubov [1937b] have initiated the study of quasi-regular points and of
the ergodicity of the measures μω = w*-\im Α* εω for continuous transformations τ in a
compact metric space K. In particular this leads to an ergodic decomposition of invariant
probabilities. Much of the early work on this can be found in the paper of Oxtoby [1952];
see also Denker-Grillenberger-Sigmund [1976] and Furstenberg [1981].
Furstenberg [1961] has studied a set of homeomorphisms of the torus including affine
transformations and he showed when all points are quasi-regular. This has led to a
considerable development; see Dani [1981] and Dani-Muralidharan [1983].
The Krylov-Bogolioubov decomposition has been extended to Markov operators by
Beboutov [1942] and sharper results have been obtained by Jamison [1965], Lloyd
[1963], and Rosenblatt [1964] for mean ergodic S. Sine [1968] obtains a portion of the
theory under weaker assumptions. Jamison's study of the sample paths has been
generalized and sharpened in the work of Jamison-Sine [1974], and Sine [1976a].
Let us mention a few relatively elementary facts on restricting S to subsets. If В is closed
and absorbing there exists an invariant probability on B. Conversely the (closed) support
of an invariant probability is absorbing. The set of extreme points of the set of invariant
probabilities μ consists the of the ergodic μ'β.
Iwanik [1981] showed that for w* mean ergodic S every minimal closed absorbing set
carries a unique invariant probability. In general this may fail, see Raimi [1964].
For a discussion of mean ergodicity and quasi-compactness we refer to the notes of
§2.2.
Sine [1975a] studied the question of mean ergodicity on the center Z, i.e., on the closure
of the union of the supports of invariant probabilities. Atalla [1974] showed: If R is
another Markov operator on X and An(S)f-+ 0 holds iff Rf= 0, then S and R induce
operators Sz, Rz on C(Z) with An(Sz)g -> Szg for all ge C(Z). For generalizations of
these results to semigroups see Sato [1979].
The book of Schaefer [1974] treats a fair amount of the spectral theory for Markov
operators S, and results extending the Perron-Frobenius theory. Eigenfunctions and
eigenvalues have also been studied by Rosenblatt [1964], and Jamison-Sine [1969]. The
set of unimodular eigenfunctions forms a group under multiplication and so do the
eigenvalues. The paper of Jamison-Sine is also of interest for its self-contained short
development of the Jacobs-Deleeuw-Glicksberg decomposition of C(K).
Rosenblatt showed that a weak flight vector for S having a weakly compact orbit
converges weakly. This and Jamison's theorem has been extended to some reducible
operators by Sine [1974]. (Jamison had pointed out that the condition of irreducibility
cannot be dropped).
Ando [1968] has studied problems on invariant measures of positive contractions in
C(K) when AT is quasi-stonean (i.e., C(K) is a conditionally σ-complete lattice). In
particular his results apply to L^. They have been generalized by Sato [1980a]. Foguel,
Horowitz [1969], and Lin [1970] have considered positive contractions in the space of
bounded continuous functions on a locally compact space, and they have obtained some
results similar to those in the theory of contractions in Lv Typical problems are the
existence of finite and σ-finite invariant measures, convergence of ratios etc. We refer to
the survey of Foguel [1971]. Sine [1973] has pointed out that the restriction of S to the
conservative part in the sense of Foguel need not be conservative.
It does not seem difficult to extend the proof of Jamisons theorem to the multiparameter
situation.
Breiman proved his strong law only for uniquely ergodic S and Jamison [1965] pointed
out that the proof extends to mean ergodic S.

186 Operators in C{K) and in Lp
Further related references: Atalla [1981], [1983], Atalla-Sine [1976], Chersi [1982],
Davies [1982], Pakula-Sine [1977], Regnier [1970], Takahashi [1971a].
§ 5.2 Contractions in Lp, (1 < ρ < oo)
1. Akcoglu's theorem. The main result in this section is the celebrated ergodic
theorem of Akcoglu [1975], which asserts the convergence a.e. of Anf= An(T)f
for positive contractions Tin Lp = Lp(f2, stf\ μ). We assume 1 < ρ < oo, and set q
= p/(p — 1). The Banach principle implies that it is enough to prove convergence
a.e. for all/in a dense subset of Lp and a dominated estimate. The first of these
steps is easy even under weaker assumptions:
Lemma 2.1. Let Τ be a power bounded linear operator in Lp. Then Anf converges
a.e. for all fin a dense subset of Lp.
Proof. Lp is the closure of the direct sum of the set of fixed points and the space
(/— T)LP, see § 2.1. If /is a fixed point of Γ, we even have Anf = f If /is of the
form g—Tg with g e Lp, we have Anf =n~l{g— Tng). Let Μ be a bound for the
norms \\Tn\\. Then
00 00 00
J Σ \Αη/\"άμ= Σ ΙΙΑ/ΙΙ2 й Σ n-"(\\g\\p + \\T"g\\py
л=1 л=1 и=1
й Σ>"ρ((Μ+ΐ)||£||ρ)ρ<α>
л = 1
00
yields Σ \Anf\p < oo. Hence Anf-* 0 a.e. α
η = ί
The dominated estimate will first be proved for positive isometries. This case is
due to A. Ionescu Tulcea [1964], but we follow the simplified proof of de la Torre
[1976] and Charn-Huen Kan [1978]:
Theorem 2.2. If Τ is apositive isometry ofLp, andf* --= supw^ x An\f\, then ||/* ||p
uq\\f\\p.
Proof. First note that Γ maps functions with disjoint supports to functions with
disjoint supports. This follows from the fact that \\g + h\\pp = \\g\\pp + ||h\\pp holds
for g,he L^ iff g and h have disjoint support.
We can assume/^ 0. Recall the notation Mnf= Max^/ ...,Anf) and
Mn(Tf)f= Мах^ДГ)/ ..., An(T')f). It suffices to prove \\MNf\\p й q\\f\\p
N
for all N. There exist disjoint subsets Ex,..., EN of Ω with MNf= Σ Ιε^ϊ/· F°r
i = l
fixed к ^ 0, the functions Тк(1Е.А^) have disjoint supports Dik,..., DNk. Hence

Contractions in Lp 187
TkMsf= Σ \DikTk{\EiAJ) = Σ \DiJkAJ = MN(Tkn, and
i=l i=l
\\Msf\\p = \\TkMsf\\p = \\Ms(Tkn\\„ (к = 0,1,2,...).
Taking the p-th power and averaging over к = 0,..., L — 1, we obtain
(2.1) \\Msf\\> й IT1 J V (ΜΝΤ*/?<1μ.
k = 0
We now apply the dominated ergodic theorem in the special case of the
translation &->& + lonZ+ with counting measure. If F is a non negative function on
Z+ with F{k) = 0 for к = N + L -1, put MFN{j) = maxt gigJV Г1 (f (/) + F(y +1)
+ ...+F(j + i- 1)). Theorem 1.6.3. yields
(2.2) Ϊμ^^/ς 2/w.
fc = 0 j=0
If, for some fixed ω e Ω, F is the function with F(/) = Tjf(p\ (OujuN + L
- 2), and F{j) = 0, (j^N + L- 1), then the left hand side of (2.2) equals
L-l N+L-2
Σ (MN Tkf)p{(o\ and the right hand side of (2.2) equals qp Σ (Tjf(co))p.
fc=0 j = 0
Applying (2.1) and integrating we obtain
\\MNf\\puqpL-^N+Z \τ'/Τάμ = ς>Σ-1{Ν + Ι,-1)\\/\\'ρ.
j=o
Now let L tend to oo. D
The proof of the dominated estimate for positive contractions in Lp will be
attained by a reduction to theorem 2.2. We first consider the special case where
(Ω, л/, μ) is a probability space, sf consists of all finite unions of disjoint sets
Il9129...9Ia with μ(4) =» mk > 0, (k = 1,..., a). Moreover, we assume || ГЦ = 1,
^ a
and that the matrix (Щ with T\u = Σ Zylj, has strictly positive entries Tiy
i=i
Using an isomorphism we may assume Ω = [0,1 [, μ = restriction of the Le-
besgue measure λ to л/, and /t = [0, rax [, I2 = [wl9 rax + m2 [,·.., /a
= [1 — ma, 1 [. If ^ denotes the σ-algebra of Borel sets, the conditional
expectation Εj is a positive contraction mapping Lp(Q, Ш, X) into Lp(Q, s/, μ). The
plan is to construct a positive isometry Q of £ρ(Ω, 8Й, Я) such that TE^f
= E^Qf holds for all/e Ερ{β). If this is done, induction yields
Tkg = E^Qkg for all ^eip(rf) and all A;^0,
and hence |MNg| S MN\g\ rg £^ΜΝ((2)|#|. Theorem 2.2 then implies
(2.3) \\MNg\\pu\\MN(Q)\g\\\Puq\\g\\P,
the desired dominated estimate for T.

188 Operators in C{K) and in Lp
The construction of Q depends on parameters mip h{j determined later. We
split each It into half open disjoint subintervals Ii} of length A (7y) = ти > О,
(J = 1,..., a). Let τ denote the null preserving map of Ω into Ω which maps
1и affinely onto Iy
Formally. If a,- and ai} are the left end points of I} and Iip and ω e Iip then τω = at
+ га, (ω — aiy)\miy Consider a positive function Α: Ω -> R, assuming a constant
value A^ > 0 on each /i>7·. We shall show that it is possible to choose miy htj in such
a way, that the operator QfipS) = Α (ω)/(τω) has the desired properties.
It follows from
116/115 = Σ f (/ρ°τ)Α^Α = Σ(ί/ρ^ΣΑ&^Κ)
that β is an isometry iff
(2.4) i^my/m-l, 0=1,.·., a)
i = l
holds. Next let us check how we can obtain TE^f= E^Qf Let f
a
= Ш;-1 J/1 j. dA be the value of i?^/on/j. X 7Jfc/is the value of TE^fonIk9and
i = l
a
this must be equal to mkl Σ J1/kl 2/^ = ™*_1 Hhki™kififor arbitrary/1?... Ja.
i = i i
Thus
(2.5) Tik = hkimki/mk, (i, fc = 1,..., α)
is necessary and sufficient. It remains to find A0·, ra0· > 0 with (2.4), (2.5), and
^jmij = mi- Note that any fe Lp(jtf) = Lp can be identified with (/l5 ...,/,),
where/ is the value of/on /f.
Lemma 2.3. ^r any fe L+p there exists a unique fe L+q with
</,/> = 11/115 = ll/IIS = ll/llpll/lle,
and this f is given by fi=fip~i.
Proof This follows from Holders inequality, together with the characterization
of equality in Holders inequality, see Hewitt-Stromberg [1969: 190-191]. α
|| ГЦ = 1 and a compactness argument implies the existence of an/with ||/||p = 1
«и^Н ТГИр = 1. Put« = |У|. Then ||u||p = 1, and 1 ^ || 7Ί#||Ρ ^ || 7y||p = 1, shows
\\u\\p = \\Tu\\p = i.
Lemma 2.4. Put ν = Tu. Then й — Τ* ν, and both и and ν have strictly positive
coordinates.

Contractions in Lp 189
Proof. |M|P = \\Tu\\p = l implies ||0||e = 1, and 1 = <u, 0> = {TuJ}
= <w, Τ*ν}. Thus jl T*u\\q й 1 yields ύ=Τ*ύ. As all Tu are strictly positive,
ν = Tu has strictly positive coordinates щ. It is simple to check that the i-th
coordinate of T*g is given by ^(т^т() Tygj. As ν has strictly positive
coordinates, also и = Τ*ύ has strictly positive coordinates. Therefore the coordinates щ
of и are strictly positive, α
The identities ν = Tu, й = T*v can be written in the form
(2.6) щ = Σ4Τβ9 mtuf-1 = Σ ^Гу!^-1.
j j
Now put hij = Vi/tij, and mi} = и^т{ 7},·/^. The first identity in (2.6) yields Zjmo'
= mt. (2.5) follows from the definition of Ay and mij9 and the second identity in
(2.6) leads to (2.4). We have completed the proof of (2.3) in the case of a finite
dimensional space L2 under the restrictions || ГЦ = 1, Ttj > 0. This will be enough
to prove:
Theorem 2.5 (Akcoglu's dominated ergodic theorem). If Τ is a positive
contraction in a space Lp with 1 <p < oo, then ||sup„^1yln|/|||p rg ?||/HP.
Proof It is simple to see that we can assume the underlying measure space
(Ω, stf\ μ) σ-finite. Then there exists a strictly positive ψ with \ψάμ = 1. Let μ' be
the probability measure with άμ'\άμ — ψ. The transformation
Lp^)3f^Vf:=flxpV*eLpW)
is a bijective order and norm preserving isomorphism of Ερ(μ) and Lp(//). Τ
= VTV~X is a positive contraction in Lp(//)? and the dominated ergodic theorem
holds for Γ if it holds for T\ since {T')k = VTk V~x. Therefore we can assume
μ(Ω) = 1 henceforth.
Assume there exists an N and/e Lp with ||MN/||p > q\\f\\p. For any ε > 0
there exists a finite sub-a-algebra 3F с &J (depending on /), with
||rl/-^rl/||p<e,(/ = l,...,Ar).Then
||Г/-(^гу^/||р<г(/+1)
holds for i = 0, and the inductive step i -* ι + 1, (i < TV), follows from
II r+1/- (^ ry+1 ^/||p ^ и г+1/- ^ r+1/||p
+ ll^r+v-(^ry+1^/||p^e + ||ry-(^r)l'^/||p?
since ϋ^- Г is a contraction in L2 {&\ Choosing ε > 0 small enough the function
MNj'will be approximated arbitrarily well in Lp-norm by MN (E& T) E^f, and we
will have \\MN(E^T)E^f\\p > q\\f\\p ^ q\\E^f\\p. This means that the
dominated estimate fails for the positive contraction E& Tin Ερ(Ω, ^, μ). It is
therefore sufficient to prove the theorem in the case when the σ-algebra sf is finite.

190 Operators in C(K) and in Lp
Now represent Γ by a matrix (7y as after the proof of theorem 2.2. Again
assume ||MNf\\p >q\\f\\p for some/^ 0. For 0 < α < 1 close enough to 1 we
have || ΜΝ(αΓ)/||ρ > q\\f\\p. Clearly ||αΓ|| < 1. For suitable β > 0 the operator
Г with Τύ = αΤν + β satisfies ||Γ|| = 1. MN(T')f^ MN(aT)f implies
ll^vi^'J/llp > 9ll/llp· But ^ is an operator which has all the properties used in
the special case treated first. We have arrived at a contradiction, α
Combining lemma 2.1, theorem 2.5 and the Banach principle we have proved:
Theorem 2.6 (Akcoglu's ergodic theorem). If Τ is a positive contraction in a space
Lp, (1 <p < oo) then Anf converges a.e.for all fe Lp.
2. Self-adjoint operators in L2. Our next aim is the dominated estimate of Stein
[1961a] for the unaveraged sequence Tnf
Theorem 2.7. If Τ is a self adjoint positive contraction in L2? an^
f***=suPnbiT*\f\9 then ||/**||2 й 6\\f\\2.
Proof We can assume/^ 0. Ρ = Τ2 is non negative definite and has a spectral
ι
resolution Ρ = J λάΕ{λ). We begin with the identity
о
Pnf- An+1 (P)f= (η + Ι)'1 Σ k(P* - Pk-')f, (η ^ 0),
fc = l
which can be easily checked. By the inequality (ax + ... + an)2 ^ η {a\ + ... + a2)
00
the square of the right hand side is bounded above by hf ·= Σ k(Pkf— Pk~1f)2.
Now | Pnf- An+1 (P)f\2 й hf implies
sup T2nfu sup An(P)f+ h)12.
00
By Akcoglu's estimate ||sup^B(P)/||2 й 2||/||2. Σ kXk~x is the derivative
n^l fc=l
(1 - λ)~2 of (1 - A)"1. Hence Σ k{Xk - A*"1)2 = (1 - λ)2 Σ Ы2{к~Х) is
fc = l fc=l
bounded by 1 for 0 rg λ ^ 1. By the spectral resolution of Ρ we obtain
py2lli = fA/^= Σ*ΙΙ^/-^"7ΙΙ
fc = l
f {Цк{Хк-Хк-1)2)а\\Ех/\\1 й \d\\Ej\\l = Ц/11
Ok 0
n2n r\\ <τθ|Ι/ΊΙ ι II /ΊΙ С^Лог-Ь? ||n1irk T^n + l ■
Thus ||supT2Y||2^2||/||2 + ||/||2. Similarly ||suPr2"+1/||2 fg 3||Г/||2
^3||/||2. g

Contractions in Lp 191
If Γ is, in addition, non negative definite, and Px fthe projection of/on the fixed
space of Γ, then the spectral theorem shows that/— i\/is the norm-limit of
fn*=E(l — i/n)f. As the norm of the restriction of Τ to E(i — l/n)L2 is
й 1 - 1/if, we find Σ* II Гк/П||2 ^ Sk(l - i/nf\\ f„||2 < oo. Hence Tkfn tends to 0
a.e. for к -* oo. It follows that the set of elements ge L2, for which Гк# converges
a.e. is dense in L2. The dominated estimate and the Banach principle then yield
Theorem 2.8. If Τ is a self-adjoint, non negative definite, positive contraction in L2,
then Tkf converges a.e. to Pi ffor all fe L2.
3. A counter-example. The positivity of T, which is not needed for von
Neumann's mean ergodic theorem, is essential for a pointwise ergodic theorem in
L2. This was recognized by Burkholder [1962a] who saw that an old example of
Menchoff [1923] in harmonic analysis could be used to construct
counterexamples in ergodic theory.
Menchoff showed the following: If (Ω, sf, μ) is [0,1] with Lebesgue measure
there exists an orthonormal basis {φη, η ^ 1} of L2 and an/0 e L2 such that the
sequence Pkf0 of projections of/0 on the subspaces spanned by {φΐ9..., <pk},
diverges a.e..
We now use this (deep) fact to show that there exists a unitary operator Tin L2,
00
for which ^„/diverges a.e.. An arbitrary fe L2 has a representation Σ am<pm?
oo m = l
where the am's are complex numbers with Σ \ocm\2 < oo.
m = l
If {Am} is an arbitrary collection of real numbers, the map
00
m = l
is a unitary operator in L2, and the powers Tj are given by
00
TJf= Σ ea"Jccm(p„
m = l
1
Ш
We now have to choose {Am} in a suitable way. Observe that σ(λ,ή)-=η ί Σ e
ι = ο
tends to 0 for fixed λ > 0 when η -^ go, and that σ(λ, ή) tends to 1 for fixed η when
λ -* 0. Take пг = ί = λΐ9 and let εχ > ε2 > ε3 > ... be a sequence tending to 0. If
λΐ9..., Як_х have been determined, find nk such that n^nk implies | <т(Дто, w) [ < εΛ
for w = 1,..., к — 1. Then find Xk > 0 with | a{Xk9 n) — 11 < sk for я <; ик. Using
11Лк<Р«112 = 1к(Аж,л,к)<рж||2<ек for /w^fc-1, and МАЯк-1)<рт\\2
= ||(<т(Ада, ик) — l)<pm||2 < 8k form ^ A:, and the orthogonality of the orbits of all
<pm, we obtain
\\a„j-(i- Pu.jfWl й *Σ ΚΙ2 ΜΛΙΙΙ
m = l
oo
+ Σ l02ll(^-fl?Jl!^ek2ll/ll!.

192 Operators in C{K) and in Lp
00
If sk tends to 0 fast enough, this yields Σ \Ankf0 — (I— Pk-i)f0\ < oo a.e..
fc = l
But then Ankf0 has, up to nullsets, the same set of divergence as (/— Pk-i)f09
and hence as Pkf0.
Notes
1. The proof of Akcoglu's theorem was the breakthrough after previous contributions by
Chacon, McGrath, and Olsen, see e.g. Olsen [1973]. Our presentation was influenced by
Derriennic [1979]. The key idea of Akcoglu was the construction of a dilation Q of T. The
oldest and best known dilation theorem, due to Sz. Nagy (see e.g. Sz. Nagy and Foia§
[1970]) asserts that if Γ is a contraction of a Hubert space §', then there exists a larger
Hubert space § = §' © ξ>" and a unitary operator (/in §, so that Tnx = PIT χ for each
η ^ 0 and * e §', where Ρ is the projection from § to §'. The main novelty of β was the
preservation of the order structure. Akcoglu's Q actually was invertible. He also
constructed order preserving dilations in Lx; [1975л]. As the existence of positive dilations seems
of independent interest we mention the following dilation theorem of Akcoglu-Sucheston
[1977].
Theorem 2.9. Let T: L -> Lbe a positive contraction on an Lp-space L,(l^p< oo). Then
there exists another Lp-space L and a positive invertible isometry T: L-> L so that D Tn^
_ ρ f*4)for alln = 0,1,2,..., where D: L -» L is a positive isometric imbedding ofL into L
and P: L -» L is a positive projection. s
Kern, Nagel, and Palm [1977] have a lattice theoretic approach to dilation theorems and
investigate which properties of Τ carry over to f; see also Nagel-Palm [1982]. Peller
[1978], [1981] obtains a dilation for operators in Lp which have a norm contracting linear
modulus, and gives applications to generalized von Neumann inequalities.
Burkholder's original example was only a contraction. Akcoglu-Sucheston [1975a]
remarked that it could be made unitary, see also Akcoglu [1979]. Akcoglu-Miller [1976]
00
have shown that for a unitary Γ in § the operator Σ ΡηΑη is bounded for every choice of
orthogonal projections Pn iff ζ = 1 is an isolated point of the spectrum. As Burkholders
example assumes/? = 2, it is unknown if ^„/converges a.e. for non positive contractions
in Lp with 1 <p<co,p φ 2. Dominated estimates in Lp under supplementary conditions
appear in de la Torre [1977], Sato [1981b], Duncan [1977]. Gaposhkin [1981a] gives
necessary and sufficient conditions for the pointwise ergodic theorem for normal
contractions in L2.
De la Torre [1978] has an extension of Akcoglu's ergodic theorem to functions/taking
values in the Banach spaces <., (1 < r < oo). Akcoglu showed that the assertion of the
Chacon-Ornstein theorem does not extend to positive invertible isometries in Lp,
(1 <p < oo). More elementary examples were given by Fong and Lin [1976].
2. Theorem 2.7 and relatives are discussed in Stein [1970]. Stein [1961a] also proved
II/**Up < Cpll/llj» with Cp < oo independent of/, 1 <p < oo, for self-adjoint Lx — L^-
contractions Τ which need not be positive; for the case ρ = 2 see Burkholder-Chow
[1961]. It turned out that it is not important to have powers of a single Tin this result.
Rota [1962] considered products Γ10 = /, Tln ··= Tn Tn^ ... Tu (n ^ 1), and proved a
dominated estimate and convergence a.e. for 7i* Tlnf, (je Lp,l<p< oo), when the Tt are bi-
stochastic. This was generalized by Doob [1963]. Finally Starr [1966] proved the
following mutual generalization of the results of Stein, Rota and Doob:

Contractions in Lp 193
Theorem 2.10. Let Tu T2,... be Ll — L^-contractions. Then ||sup„^01 Tfn Tlnf\ \\p
= Я \\f Upholds for 1 <p < oo. Forfe L1 with J|/| log+ \ί\άμ < oo, and he Ll with^ ^ h
^ Min(|/|, U) we have
n^o e — ι
If the Tt are also positive, then TfnTlnf converges a.e. for f belonging to some Lp,
(1йР< oo), with Jl/l log+ |/|</μ < oo.
Starr [1965] showed that there is no such result for the reversed products 7i T2... Tn
instead of Tln. Burkholder [1962] showed that the theorem is false if/belongs to Lx only,
see also Ornstein [1968]. For related material; see Rao [1979].
3. Lin [1974d] studied unaveraged convergence in Lp for Markov operators: If Γ is a
conservative and ergodic positive contraction in L1 with σ-finite invariant measure ν and
Г*п1л-»0 a.e. holds for some A with 0<v(A), then || T*nf\\p -> 0 for /e Lp,
(1 <p < oo). || T*nf\\p -► 0 for all/e Lp holds if the past remote σ-algebra in the
corresponding bilateral product space has no sets of nonzero finite measure.
4. The following theorem follows from results of Sz. Nagy and Foias [1960], see also
Foguel [1963], [1969]:
Theorem 2.11. If Τ is a positive contraction in L2 the set
*:= {/e L2: || Tnf\\2 = || Γ*"/||2 = ||/||2 for all η ^ 1}
is a closed linear subspace invariant under Τ and T*, and on К one has TT* = T*T= I. For
f±K, w-lim Tnf= w-\im T*nf= 0.
5. Mme Benozene [1981] proved the convergence a.e. of averages
n-l
n-i ^] I Tjf\a sign (Tjf) for Τ a positive contraction in Lp, (1 <p < oo), fe Lp, and
\ <a<p. Such sums are of interest, when one looks at averages of a type introduced by
n-l
Beauzamy and Enflo, namely the elements s(np)(f) minimizing n~l Σ \\g — Tjf\\p. (In the
j=o
case of Hubert spaces or ρ = 2, one is led back to usual Cesaro averages). Benozene studies
the convergence of s„p)(/) also for nonlinear T; see also Guerre [1978].
6. The proof of Akcoglu's theorem for power bounded positive Τ is an open problem.
Brunei and Emilion [1984] showed that the power bounded case would imply even the
Cesaro bounded case. Feder [1981a] has constructed a (non positive) power bounded Τ
with power bounded inverse in Lp, (1 <p ^ 2) for which ^4„/does not converge a.e..
Assani [1985] has examples for 1 <p< oo. Kan [1978], [1983] has positive results if Τ is a
"Lamperti operator" (maps functions with disjoint supports to functions with disjoint
supports). Sato [1974], and Fong-Lin [1976] consider the generalization of the Sucheston
decomposition of Ω into the remaining and disappearing parts. Other partial results have
been obtained by Assani [1984] and Emilion [1984a]. Assani and Mesiar [1984] studied
the a.e.-convergence of n~aTnf for power bounded T.
7. Hachem [1982] proved the convergence a.e. and in Lp-novm of n'1 Fn, when (Fn) с Lp is
a superadditive process for a positive contraction Τ in Lp, (1 <p<co), and
lim infUw"1 Σ (Д- TF^)||p <oo.
i=l

Chapter 6: Pointwise ergodic theorems for
multiparameter and amenable semigroups
Some multiparameter pointwise ergodic theorems can be deduced by a simple
inductive argument from the 1-parameter case. This is done in the first section.
Section 2 treats the principal pointwise ergodic theorems for multiparameter
additive and subadditive processes, and section 3 the maximal ergodic theorem
and the pointwise ergodic theorem for multiparameter semigroups of
contractions in Lt and integrable functions. The chapter ends with a discussion of
ergodic theorems for amenable semigroups.
§6.1 Unrestricted convergence for averages over (/-dimensional
intervals
Let d ^ 1 be a fixed integer and let V = {0,1, 2,... }d be the additive semigroup of
d-dimensional vectors with non negative integer coordinates. For и = (щ),
ν = (Vi) e V we write и rg ν if щ ; rg vb (i = 1,.., d), и < ν if щ < vh (i = 1,..., d),
and [u,v\_= {w eV'.u^w <v). If A is a finite set card (A) is the number of
elements of A. For η = (nl9 ...,nd)eV let
d
π(η) — Π ην = cai*d ([0, n[)
v = l
where 0 = (0, 0,..., 0) is the neutral element in V.
For neV and operators 7\?..., Td we shall use the notation
7-1 'T'fli rT4l2 T^d
η — *1 22 - λά
\= Σμ€[0,π[ Tu
An= niny1 Sn.
η -> oo shall mean that nv tends to infinity for ν = 1,..., d.
If Anf converges a.e. as η -* oo we say that we have unrestricted a.e.-
convergence. Unfortunately unrestricted a.e.-convergence does not always hold
for/e Lx even if the T{ are given by commuting measure preserving
transformations τ1?..., τά. In this section we study unrestricted convergence. In the
subsequent sections we will obtain convergence for all/e Lt under some restrictions,
e.g. by taking averages only over squares.

196 Multiparameter and amenable semigroups
Theorem 1.1 (Zygmund-Fava). Let Tu ...,Td be Lx — L ^-contractions in Lx of a
finite measure space. Then limn_00 Anh exists a.e.for he L logd_1 L, and it equals
Α^(ΤΧ)A^(T2)...A^Td)h, where А^(Тд/=Шп^^Ащ(Тд/.
Proof. We first assume that the 7J are positive. The case d = 1 is theorem 1.7.3.
For the induction put/fc = Ak(Td+i)h. For he L logdL, theorem 1.6.4 implies
sup|/J g L logd_1L. Apply theorem 1.7.5 with L = L logd_1L, Л = V, Τλ =
= An, λ = η = (nl9..., nd), к = nd+1. The assertion follows from
Anfk = Ani(T1)...And+i{Td+1)h.
In the general case let Tt be the linear modulus of 7]. It is again an Lx — L^-
contraction, and, hence, an L2-contraction. It follows that T{ is an L2-
contraction. Now note that theorem 1.7.3 remains true for non positive Г if the
maximal estimate is done with | T\. Also theorem 1.6.4 holds for non positive
operators because of \An(T)f\ rg \An{\ 7Ί)|/||. Similarly, the argument in
theorem 1.7.5 goes through with Τλ = An by using >4ni(T1)... And(Td) instead of
Τλ in the proof. It follows that theorem 1.1 remains true with the same proof, d
Theorem 1.2. Let 7j,..., Td be positive contractions in Lp, (1 <p < oo). (The
measure may be σ-finite now). Anh converges a.e. for η -> oo and h e Lp, and
\\sup{An\h\:neV}\\pu(p/p-m\h\\p.
Proof. Put hd+1 = \h\ and ht = sup {Am(T^hi+1: ще U\, \^i<Ld. Akcoglu's
dominated estimate implies ||ht\\p ^ (ρ/ρ — 1)||Λί+1 ||ρ. Therefore, the second
assertion follows from sup {An\h\: η eV} = hx. The convergence of Anh is
Akcoglu's ergodic theorem for d = 1, and it follows for general dby the inductive
argument in the previous proof, α
Theorem 1.1 was proved by Zygmund [1951] for the case of measure preserving
transformations (and continuous time) and by Fava [1972] for operators. See
also Dunford [1951] for the case d = 2. The simple argument with theorem 1.7.5
is due to Sucheston [1983]. The multiparameter form of Akcoglu's theorem was
proved by McGrath [1980a]; see also Olsen [1983].
The integrability condition he L logd-1 L is in a sense weakest possible for
unrestricted a.e.-convergence: To see this consider a family {ZM, и е V} of
independent identically distributed random variables. In the product space
representation of multiparameter stationary processes, similar to that in § 1.4, we
can assume that Xu is the w-th coordinate variable in Ω = Ry. If τΜ is the shift
defined by Χν(τ„ω) = Xu+V(a>), we can write Xu = Χ0°τη. We obtain AnX0
= π{η)~χ ΣΜε[ο,η[^Μ· By theorem 1.1, X0e L logd_1L implies the convergence
a.e. of AnX0. Smythe [1973] proved that, conversely, the unrestricted
convergence a.e. of π(η)'χ ΣΜ6[ο,η[^Μ implies X0 e L logd_1L when the Xu are i.i.d..
Infinite μ. We now turn to the infinite measure space result analogous to theorem

Unrestricted convergence 197
1.1. (It will not be needed in the rest of the book). For infinite μ, L\ogd~lL is
clearly not the proper class of functions for the proof of a. e. convergence of Anf
because it contains all/with \f\ <; 1. Fava [1972] introduced, for к = 0,1, 2,...,
the class 9?k of all measurable functions /: Ω -+ R for which
(1.1) ί ^(Ю.^)'*
i\f\>t) l \ l J
is finite for all / > 0. Clearly, for finite μ, this class coincides with L logkL. For
infinite μ and к ^ 1,9?k is a proper subclass of L logkL. Obviously Lx is contained
in 9l0 and, for 1 < ρ < oo, Lp is contained in each 9?k. It is fairly easy to check that
each 9?k is a vector space and that among the classes 9?k we have the inclusions
We shall now present Fava's proof of unrestricted a.e. convergence for/e 9?d-!.
Let Lx + L^ be the space of all measurable functions/: Ω -* R, which can be
written as a sum/ = /1 +/2 with/i eL1,f2eLao. We know from § 1.6. that a
positive Lx-contraction extends uniquely to the space of non negative
measurable functions and in particular to (Lx + L00)+. If Γ is a positive Lx — L^-
contraction this extension maps (Lx + L^y into itself in a linear way and
therefore uniquely determines an extension to Lx + L^ also denoted by Γ, and linear
in Li + L^.
Definition 1.3. An operator Μ mapping Lx + L^ into the space of real valued
measurable functions is called an abstract maximal operator if it has the following
properties:
(i) /^ 0 => Mf^ 0 (Positivity);
(ii) \M(f+g)\u\Mf\ + \Mg\; \M(af)\ = \a\\Mf\ (fge L± + L^aeU)
(Sublinearity);
(iii) 0ufug=> MfuMg;
(iv) WMflLuUlLAfeL^LJ;
(ν) Μ is of weak type (1,1), i.e., there exists о 0 with λ > 0, fe Lx =>
By lemma 1.6.1 the operators M^ = Μ^(Γ) withM^/= ΜΝ(Γ) |/| are abstract
maximal operators, (Ne Ν υ {oo}), when Г is a positive Lx — L^-contraction.
We shall only deal with finitely many abstract maximal operators and may use
the same constant о 0 for them. For fe Lx + L^ and / > 0 we shall use the
convenient notation f =fi{f>t)9ft =Я{/^} =/-/*·
Lemma 1.4. If Μ is an abstract maximal operator, fe (Lx + L^)+ and t > 0, /Леи

198 Multiparameter and amenable semigroups
Proof. We may assume fx e Lx j' = p+f^Lfx + t implies
μ(Μ/>2ή^μ(ΜΓ>ί)^€Γ1\\/%. π
Lemma 1.5. For measurable/ί> Ο and Ее si
$/άμ = ]μ(Εη{/>ή)ώ.
Ε О
Proof. Put g =flE and G = {(ω, /)· t < g(co)}. Then
00 00
\%άμ = J J 1σ(ω? i)dt μ{άώ) — \ \ 1σ(ω> t) μ(άω)άί. D
β ο ο β
The main step is the following:
Theorem 1.6. Let Ml9 Ml9 ...,Mk be abstract maximal operators.
(i) For any fe 9?k, (к ^ 1), the functions M} · Mj-X ... Mxf (1 rgy ^ A:) belong
to Lx + L^. /и particular МкМк-х ... Mxf is well defined for fe 9?fc_i;
f//J There exists a constant Ck > 0 swc/z ί/ζα/
(1.2) /i(Mk...M1/>40^Q J ;flog{Y 'φ
{/>ί} t \ tj
holds for all поп negative fe 9ik-x and all t>0.
Proof. The proof is by induction on k. Because ofL1 + L00=5 9?0z59?1=D...,
M1f is always well defined.
Note that a measurable function belongs to Lx + L^ if and only if it is inte-
grable over every set of finite measure.
For к = 1, (1.2) follows from Lemma 1.4. To prove (i) for к = 1 take some
/e 91^ and some set Ε of finite measure. Then
00 00
\MJdp = J μ(Εη{Μχ/> t})dt й 4/i(Я) + J μ(Μχ/> i)dt
Ε 0 4
00 00
^ 4μ(£) + J μ(Μι/> As)Ads й Αμ{Ε) + Ac J J s~l/άμάε
^ 4μ(£) + 4c i/l{/gl) j ί_1ί/ίί/μ
Ω 1
= 4//(£) + 4ci/(log+/)^<oo.
As Ε was arbitrary with μ(£") < oo we see that Mxf belongs to Lx + L^.
Now assume that the theorem has been proved for к operators and consider к + 1
abstract maximal operators Ml9...,Mk+i. We first prove (ii) with к replaced by

Unrestricted convergence 199
к + 1. Take/e 9?fc+. By (i) of the induction hypothesis Mk+iMk... Mxf is well
defined. Using
Mk+1...MJ^Mk+l...MJ2t + 2t
and Lemma 1.4 we obtain
μ(Μ1ί+1...Μί/>4ή^μ(Μ,+1...Μ1/2ί>2ή
й- J Μ1ί...Μ1/24μ
t {Mk...Mif2t>t}
= c J Μ^,.,Μ^άμ withg= t~lf2t.
{Mk...Mig>l}
Put ρ(s) = μ(Μλ ... Mxg > s). By Lemma 1.5 the last integral equals
00 00
(1.3) ^({Mk...Mlg>\}n{Mk...Mlg>s})ds = Q{\)+lQ(s)ds.
0 1
As 9?k is contained in 9?k_1? g belongs to 9lk-! and the induction hypothesis gives
us
(1.4) Q{s)u4Ch f ^flog^Y V
{A.g>s} S \ S J
This can now be used to estimate the right hand side of (1.3). We obtain
00 oo / Лд\к-1
(1.5) ^Q(s)dsuU4Cks-1 J g log-M άμ-jds
1 1 {4-g>s) \ S J
*9 ( 4p-Y-1
= 4CtJ[gJ log-5- i-1^]^
Ω 1 V * /
Applying the elementary inequalities log+ab rg log+a + log+6 and (a + b)k
^ 2k(ak + 6k) for a, b ^ 0 we see that there are constants Ak9 Bk>0 such that the
last term is
^Ai^ + 4is0og+g)*^
{f>2t} t {f>2t) t \ t/
^(Λ + 4) J {(log+{)V
Similarly (1.4) yields

200 Multiparameter and amenable semigroups
в(1)^4Ск f g(log4g)k-1dμ = 4Ck$g(\og+4g)k-1dμ
{4flf>l} Ω
ύΑί^άμ + Βί^(Ιοζ^γ-4μ
йМ + ва f {(W)V
{/>t} t \ t)
Combining our estimates we get (1.2) with к replaced by к + 1 und with Ck+i
= c(Ak + Bk + A'k + W.
To prove (i) with к + 1 operators consider fe 9lk + x and a set Ε of finite
measure. By Lemma 1.5 and by what we have proved above
00
J Mk+i... Μί/άμ = J μ(Εη {Mk+1... MJ> t})dt
Ε 0
й 4μ(Ε) + ] μ{Μκ+1... MJ> t}dt
4
<HM£) + Q+J[ J ^Υΐο8^Υ<//ι]Λ.
4 {4/>ί} ϊ \ ϊ /
The finiteness of this expression follows from/e 9?k+1 c: L logk+1 L by the same
computation as in (1.5). D
For η = (nu л2,..., nd) put Mif= sup {^„/: u ^ л}, and let e = (1,1,..., 1).
Then theorem 1.6 and the estimate Mdn\f\ S Μηχ(Τγ) МП2(Т2)... Mnd(Td)\f\
yield
Corollary 1.7. Le/ Г1?..., Td be positive Lx — L ^-contractions. There exists a
constant Cd > 0 such that
/i(M-J/|>40^Q f Wlog^fY '*
holds for all t>0 and all fe 9?d_i.
Theorem 1.8 (Zygmund-Fava). Let Tl9..., Td be positive Lx — L ^-contractions,
then Anf converges a.e. as η -> oo for all fe 9^-!·
Proof. We may assume/^ 0. Put
Λ (g) = lim sup Ang - lim inf ^Bg.
n->oo n->oo
By theorem 1.2 zl (g) is 0 for g e Lp? (1 < /? < oo). Let/k, (A: ^ 1) be an increasing
sequence in L+ with/k ->/a.e.. Then
j (Л й δ (/-Λ) + zi (Λ) = zi (/-Λ) ^ MUf-Λ) ■
For any fixed t > 0,

Multiparameter subadditive processes 201
j ^(logb/sY"1^
if-fk>t) t \ t J
tends to 0 as к -> oo, since the integrands tend to 0 a.e. and are bounded by the
integrable function l{/>i}/"1/(log(///))d_1. Therefore Corollary 1.7 implies
л m = ο π
Л(/) = 0. D
Notes
For the continuous parameter form of the Zygmund-Fava theorem the reader is refered to
Fava's paper and to Sato [1981a]. Sato also has further related estimates for maximal
operators.
Yoshimoto [1979] derived dominated estimates for vector valued functions in $Kfc.
Yoshimoto [1982] also studied similar estimates for Lp — L^-contractions.
Converse maximal estimates were given by Dang-Ngpc [1980] for groups and by Sato
[1983 a] for semigroups.
§ 6.2 Multiparameter additive and subadditive processes
1. Processes indexed by intervals. We now consider d ^ 1 commuting endomor-
phisms τ1? τ2?..., τά of a measure space (Ω, sf9 μ). We continue to use the
notation given in the beginning of the previous section, but now 7] will always
be the operator/-» 7]/ = /° τ£ in Lx. We shall prove the a.e.-convergence of Anf
= π (и)-1 Snf"for all/e L1? when η tends to infinity along an increasing sequence
in V = {0,1, 2,... }d. More generally, we shall obtain an analogous result when
the additive process Snf is replaced by a subadditive process.
The commuting endomorphisms τ{ define a semigroup τ = {τΜ: ueV} = {τΜ}
by τΜ = τ^1τ"22... τ^, where и = (uu ..., ud). Let β be the set of non empty
intervals [w, t;[, (w, ν e V). For 7 = [w, t;[, 7+ w is the interval [u + w, ν + w[. Put
e = (l,l,..-,l).
Definition 2.1. A superadditive process (with respect to τ) is a set function
F: f э1-+ FjE Lx with the following properties:
(2.1) FIo%u = FI+U whenever Ie f and we V;
η
(2.2) if Il9 ...9In are disjoint sets in f and if I — (J It is also in β, then
i=l
i = l
(2.3) y(F):=supL^f№/e/}<oo.

202 Multiparameter and amenable semigroups
We may write Fin the form F= {Fj} = {Fj: Ie /}. If we want to emphasize
that τ is a discrete semigroup we call Fa discrete superadditive process. y(F) is
called the spatial constant of F.
If — F is superadditive then F is called subadditive. In the definition of the
spatial constant one must then write inf instead of sup. If both F and — F are
superadditive then Fis called additive. As the semigroup τ was assumed discrete,
the additive processes are just those of the form FT = ΣΜε/ Tjf with an/e Lx.
Let us call F2-superadditive if (2Л) and (2.3) hold and (2.2) holds for η = 2. For
d = 1 superadditivity and 2-superadditivity coincide.
As in the case J = 1, the definition of a superadditive process can be given in an
equivalent way without using endomorphisms. (2.1) is then replaced by the
condition that for any finite collection (71?..., Ik) in β and for any и е V the joint
distribution of (FIl9..., FIk) is the same as that of(FIl+u,...,FIk+u). We shall use
the notation FI = card (I)~1FI.
Let us look at some examples of two-dimensional subadditive processes:
(a) Assume that random straight lines are given in the plane in a stationary
way. E.g. their perpendicular distances from the origin are the points of a
homogeneous Poisson process on R+ and their directions are uniformly and
independently distributed in [0, 2π[. For u,veV let R(u, v) = {te R2: и ^ / < v}. (As
[w, ν [ contains only points from V it differs from R (u, v).) For / = [w, ν [ let FI be
the number of lines intersecting R(u, v).
(b) (Cluster process of Grimmett [1976]). Let Xu, (ue V), be independent
random variables taking the value 1 with probability/?, (0 <p < 1). We connect
two elements u,ve\l with Euclidian distance 1 if Xu = Xv = 1. Thus, within each
/ g f we have a random graph. Let Fj be the number of connected components of
this graph. Fj is called the number of clusters in /.
The papers of Hammersley-Welsh [1965], Hammersley [1974], Smythe
[1976], and Nguyen [1979] contain other interesting examples.
A real valued set function g: f -* R is called 2-superadditive if g{A) + g(B)
-ug(A\jB) holds for any disjoint A.Bef with A\jBef, and
g(A) = g(A + u) holds for any Ae β and ueV. The proof of the following
Lemma is an easy exercise similar to that of Lemma 1.5.1.
Lemma 2.2. For any 2-superadditive g, 7i(n)~1g([0, n[) converges to
sup {card(I)~ig(I):Ie f} e Ru{oo} for η ->οο.
Clearly the set function g defined by g(I) = ^Ιάμ is 2-superadditive if F is 2-
superadditive, and (2.3) then guarantees that the limit is finite. We see that the
spatial constant both of 2-superadditive and of 2-subadditive processes can be
written as
(2.4) y(F) = lim тфГЧ^ф = lim }Ρί0Μ(Ιμ.

Multiparameter subadditive processes 203
Let J denote the σ-algebra of all sets A e si^ which are invariant under all τί5
(ι = 1,..., d). As usual the ergodicity of τ means that J is trivial.
As its proof is similar to the 1-parameter case we only state the mean ergodic
theorem for F due to Smythe [1976]:
Theorem 2.3. If F is a 2-subadditive process in a finite measure space, then F{0n{
converges in Lx-norm as η -> oo to an J-measurable fi For any A e «/,
J/<//i= lim $Ρ[0Μάμ.
A n-*co A
In particular, for ergodic τ,/= /i(Q)~1y(/r).
We now turn to the question of a.e.-convergence. The first multiparameter point-
wise ergodic theorems go back to Wiener [1939], who considered averages of
functions/0 τΜ with и ranging over spheres in Rd. His ideas will play an important
role in what follows.
We shall prove that FIr converges a.e. for any subadditive process if/r tends to
V in a regular way. This was shown by Akcoglu-Krengel [1981] and we follow
their arguments.
Definition 2.4. A family {/r}r 6 N (or a finite such family) is called regular (with the
constant С < oo) if there exists an increasing sequence I\ c/jcin/ with /, <= I'r
for all r such that card (Гг) ^ С card (/r) holds for all r.
If {Q can be chosen in such a way that the union of the sets I'r is V, we shall
write lim,.^/,. = V.
Obviously any increasing sequence of sets in f is regular with the constant 1. The
notion of a regular sequence in f also allows to treat another interesting
example: We say that a sequence η (к) remains in a sector of V if there is a finite C0
such that the ratios ni(k)/nj(k) are bounded by C0 for 1 ^ ij' ^ d and all k. If
л (1), и (2),... is a sequence in V remaining in a sector and tending to infinity we
may find a permutation m(l), m(2),... of it for which Max {т{(к)\ 1 ^ / ^ d] is
increasing. The sequence [0, m(r)[ in f is regular with constant Cq. Therefore
the a.e.-convergence of FIr for regular sequences tending to V will imply that
F[o,n(k)[ converges a.e. for sequences n{k) -* oo remaining in a sector of V. The
definition of a regular sequence /l5 /2,... also allows that the lower corner of the
intervals tends to infinity.
In the next lemma we write — Q + C2 for {v: 3we Q with w + ve C2}.
Lemma 2.5. Let I[ cz Г2 cz ... I'N be finitely many nested sets in β'. Let A be a finite
subset of У and let k:u -> k(u) be a map of A into {1,..., N}. Then there exists a
finite subset A' a A for which the sets Du := /jL·) + u,(ue A'), are disjoint, and for
which A is contained in the union of the sets DM := — Vk(u) + (Ι£(Μ) + u), (и в A'). In
particular, card (A) :§ 2d ΣηεΑ' саг<^ (/ад)·

204 Multiparameter and amenable semigroups
Proof. Let JiN be a maximal collection of disjoint sets of the form ΓΝ + и with
к (и) = N. When JfN, ...9Л{ have been constructed for some i > 1, let M{ _x be a
maximal collection of sets of the form Du = Ifi-1 + u with к (и) = i — 1, for which
all sets in iHu... υ<ν#Ν are disjoint. Let A be the set of all ueA with
Due JjU... uJfN. Take any t; e ^4. By the maximality of ^fc(l7) there exists
some j ^ k(v) and Д, e ^ with DOnDu + 0. Then fc(w) ^ A:(t;) and /к(м) =э Гт.
Now (7к(м) + f) η (7k(M) + u) φ 0 yields t; e Д. The last assertion follows from the
inequality card (—Ik + (Ik + u)) ^ 2d card (Ik). π
Theorem 2.6. Let {Ix, I2,..., /#} 6e a finite regular family in f with constant C.
Let Bu B2 be other sets in f such that B2 contains all sets 2?х + Ik —
= {v + weV:ve Bx,we /k}? (k = 1,..., N). Let F = {Fj} be a non negative su-
peradditive process and define
E = {ω e Ω: max FIk(a>) ^ a}
with a > 0. Then
(2.5) ^-^XDM^^IF.^
holds for any D e si.
Proof For each ω e Ω put Α (ω) = {ue Bx: τηω e E}. For any De si
(2.6) Jcard(^))/^CO) = J Σ 1{ω:τΜωβ£}Φ= Σ μ(Ώ ПТ~Х Е)
D DueBi ueBi
^ Σ [μ(τ,Γ1£)-μ(Ω\Ζ))] = οαΓ(1(51)[μ(£)-μ(Ω\β)].
U (Ξ Β χ
For each w e Α (ω) there is an integer k{u), 1 й k(u) ^ N, with
(2.7) F/k(M)(TMco) = Fu+Ik(u)(co) ^ a card (/k(M)).
As the family {Iu I2,..., IN} is regular there exists a nested family
I[ α Γ2α ...α ΓΝ in f with /k с /k and card (7k) ^ С card (7k). We may apply
Lemma 2.5 to the set A = Α (ω) and find finitely many elements w1, w2,..., ul in
ι
Α (ω) such that the sets и1 + /k(Mo are disjoint, and such that 2d Σ card (7k(Mz))
i = l
^ card (yl (ω)). Then the sets ul + /k(Mi)? (/ = 1,...,/) are also disjoint and we have
2dC Σ card (Ik{ui)) ^ card (Λ(ω)).
i = l
As all these sets are contained in B2 the nonnegativity and superadditivity of the
process together with (2.7) imply

Multiparameter subadditive processes 205
^в» ^ Σ /£<+W») ^
i=l
I
α
^Σ card (/к(в0) ^ -^ card (Λ (ω)).
Integrating this inequality over D and using (2.6) we obtain (2.5). α
Here we need only
Corollary 2.7. Let {Iu I2,...} be a finite or infinite regular family in f with
constant C, and let F — {Fj} be a non negative super additive process. For any α > 0 the
measure of the set
Ε = { ωβ Ω: sup FIk(co) ^ α}
к
satisfies
α
Proof We may assume that the sequence /1? /2? · · · is finite. If B2 is the smallest
element of β containing all sets Bx + /k? (k = 1,..., N) and if B1 in f is
sufficiently large, then the ratio card (2?2)/card (Bx) is arbitrarily close to 1. Taking D
= Ω, the estimate
μ(Ε) < 2 5L4 f ¥ΒΛμ < 2 ^/n, card (B2) y(F)
^v ' ~ α card (Bt) b <* card (5X) v 2) r v '
therefore yields the desired maximal inequality, α
To prove the ergodic theorem we must first study the additive case.
uel
is an additive process for integrable/.
Theorem 2.8 (Tempel'man [1972]). Ifxu τ2?..., τά are commuting endomorph-
isms of a measure space (Ω, «я/, μ) and if I u I2, ...isa regular family in β tending to
V, then the sequence A(r9f) = card (/r)_1 SIrf converges a.e. for all fe Lp,
(1 ^/><oo).
Proof Put
A (/) = lim sup A (r,f) - lim MA (r,f).
r->oo r->oo
Lemma 1.1.3 implies that any fe Lp may be written as a sum/ = Σ /} with

206 Multiparameter and amenable semigroups
functions/) having the following properties: For 1 <,j <; d there exists a^-ei^
with fj = gj — gj ο τ ρ fd+i belongs to L2 and is τ,-invariant for j — 1,..., d;
||/, + 2|μ<ε, and \\fd+3\\1<s2.
If ej is the y'-th unit vector in V the sequence
f/rJ = card (7,)"1 [card (7,A(Ir + e*))]
tends to 0 for all j since Ir tends to V. Therefore A (fj) = 0, (j = 1,..., d) is a
consequence of the estimate \A(r9fj)\ ^ 2||g)||00i/rj. J(/d+1) = 0 is obvious. It is
also clear that A(fd+2) is bounded by 2||/d + 2lloo < 2ε.
Because of \8Ι\/ά + 2)\άμ = card (/)||/^+3||1? the spatial constant of the
additive process {57|/d + 3|} is y({SI\fd+z\}) = \\fd + z\\v By A(fd+3)
rg 2 sup,. yl(>, | fd + 31) the set {J(/d + 3) > ε} is contained in
{supryl(r? |/d+3|) > ε/2}. Corollary 2.7 therefore gives us
2dT2
/ι(4(/, + 3)>β)£ β2.
ε
d + 3
As ε > 0 was arbitrary A (f) ^ Σ J (/}) yields J (/) = 0. α
Remark. The limit/of the sequence A(r,f) does not depend on the sequence
{Ir}. We may identify it as follows, leaving the details as an exercise: We may
assume that μ is σ-finite. By an exhaustion argument we can find a maximal set С
such that there exists a finite measure ν <ζ μ invariant under all τ} with strictly
positive density h = dv/άμ on C. The argument given above shows that/is
arbitrarily close to a function which is invariant under all τ,- and so /must be
invariant, too. If/is in Uu an application of Fatou's lemma shows \\f\\i й ll/lli-
Thus, {/> 0} must be contained in C. As any/e L£ may be written as/ = /g +/'
with l/'l < ε and/ e Lx se see that/vanishes in Cc for all/e Lp, (1 rg ρ < go).
As the sets {h ^ 1}, {1 > h ^ 1/2}, {1/2 > A ^ 1/3},... belong to J and
partition C, we may determine/on these sets separately. Restricted to such a set μ is
finite and/integrable and by the usual Lx-convergence argument/is the unique
«/-measurable function with \A /άμ = \A /άμ for all Ae J.
We are now ready to complete the proof of the following multiparameter
subadditive ergodic theorem of Akcoglu-Krengel:
Theorem 2.9. If F= {Fj\ Ie f) is a subadditive process and Il9I29... a regular
family in β tending to V, then the sequence FIr converges a.e. as r -> oo.
Proof The theorem is stated in the subadditive case because that seems to be the
most frequent case of application. But for the proof we assume (passing to — F)
that F is superadd!tive.

Multiparameter subadditive processes 207
The process {SjF[0te[} is additive and the process {Fj — SjF[0e^ is non
negative and superadditive. As Tempelman's theorem takes care of the additive case,
we may henceforth assume that F is non negative and superadditive.
Let /,. be the interval [t>(r)? w(r)[ and Tr = [y'(r), w'(r)[. Replacing С by a
bigger constant (also denoted by C) we may enlarge the sets I'r and assume v'{r)
= 0, and that for any meN there exists an L(m) such that for r ^ L{m) all
coordinates of w'{r) are divisible by m.
Let V(w) = mV = {(mil9..., mid)\ (il9..., id) e V} and let /(w) be the set of non
empty intervals [w, ν [ with щ ν e V(m).
For r ^ L{m) let Ir m be the largest element of /{m) contained in Ir and let Jr m
be the smallest element of f{m) containing /,.. (If L(m) is large enough, Ir contains
at least one non empty ^/^-interval.) As the length of the sides of Ir tends to
infinity the ratio card (/г>да)/card (^ m) tends to 1. The sequence
•••Л,т>4,т>Л+1,т>4+1,т> · · · ·> (r = L(m)) is regular, as may be seen by
comparing these intervals with .../,, Tr> I'r+1,1'r+1, We call it £m.
Let/and/denote the lim inf and lim sup of the sequence FIr, and let/m and/m be
the lim inf and lim sup of FI as I ranges through £m. Comparing FIr with FJr m and
using card (Д)/card(4,m) -* 1 we obtain fufm· Similarly, a comparison with
FIr m implies/^fm. The finiteness of/m and hence that of/,/and/m follows from
Corollary 2.7 when we apply it to £m and let α tend to infinity.
As the set {ω:/(ω) —/(ω) > α} is contained in the sets
Em,* = {®:/ж(ш) -/„(ω) > α}
it is now enough to show that μ(Ετηα) is small for fixed α > 0 when m is large.
Let ε > 0 be a given number. Lemma 2.2 shows that there is an integer m with
iF[0tme[^>y(F)-8.
We may define an additive process on f{m) by
Щ= Σ Fl0ime[otu, (/£/<">).
«6/n V<™>
Subtracting this process from the restriction of Fto f{m) we get a non negative
superadditive process Fm = {F™} on f{m). As the spatial constant of {H™} equals
^Ρ[0τηβ[άμ the choice of m implies y(Fm) < s.
As the sequence Cm is regular in f{m) and {Щ1} additive, the sequence Щ
converges a.e. as I ranges through £m. But then/m — fm must be bounded by
sup {Ff: Ie £m}. Now corollary 2.7 yields
As ε > 0 was arbitrary the proof is complete, α
2. Extensions to convex sets and to continuous parameters. We now sketch some
generalizations of results given above. The intervals will be replaced by convex
sets.

208 Multiparameter and amenable semigroups
The case of norm convergence is quite simple for additive processes. Let <ШЪ
denote the family of bounded Borel sets A with positive Lebesgue measure λ (A).
Let 2Γ = {Tu: и в R+d} be a bounded semigroup of linear operators in a Banach
space X. An additive process for У and 3&ъ is a map F: &b э A -* FA e X with the
properties:
(2.8) for disjoint AuA2e@\ FAlKjAl = FAl + FA2;
(2.9) TuFA = FA+u,(ueU+d,Ae@b);
(2.10) there exists a constant y0 < oo with \\FA\\<Z y0KA)\ (A e @b).
In V, intervals consisted only of lattice points, but, in U + d, [w, v\_ is the set
{we U+d: и ^ w < v}. Let us say that a sequence (Ап)пеЫ of Borel sets in U+d with
λ (An) -> oo consists asymptotically of unions of large intervals if for any m e N and
any ε > 0 there is an N(m, ε) such that for η ^ N(m, ε) there exists a set ^4™,ε
which is a disjoint union of translates of the interval [0, me[ and for which
λ(ΑηΑΑ™ε) < ελ(Αη) holds.
For any A cz U+d we denote by Am the union of all intervals [w, u + me[ with
w g V(m) which intersect A9 and by 4m the union of all intervals [w, и + me[ with
w g V(m) which are contained in A.
Theorem 2.10. In the situation just described assume that m~dF[0me[ converges
strongly to some J e X as m -> oo. Then λ(Αη)~χ FAn converges strongly to J for any
sequence (An) in &b with λ(Αη) -> oo which consists asymptotically of unions of
large intervals.
(The condition (2.10) may be replaced by the weaker assumption
sup {\\FA\\: Ae &b, A a [0, e[} < oo if λ(Αη) tends to oo for η -* oo and
λ(Α™\Α™)/λ(Αη) tends to 0 for each me N.)
The proof of this theorem, which generalizes a result of Fritz [1970], is
straightforward and is left to the reader. It is clear that a similar theorem holds
when U+d is replaced by V, Ud or Zd.
We now turn to the more subtle question of a.e.-convergence of sequences
A(An)~1FAn. Tempel'man [1972] has given a very thorough treatment to this
problem applying, e.g., to discrete additive processes and to integrals FA
= §AfOTua(du) even when / takes values in a Banach space. He has given
axiomatic properties for the sequence (An); see § 6.4. We consider only convex sets
An and their restrictions to V, but we treat subadditive processes.
For any Borel set A in Rd, ρ (A) denotes the supremum of all r ^ 0 for which
there exists a Euclidian sphere S(x, r) = {ue Rd: \\ и — x\\ rg r} contained in A.
A + A' is the set {x + у: х е А, у е A'}.
We need the following simple lemma of Day [1942]:

Multiparameter subadditive processes 209
Lemma 2.11. If a convex set Ε in Ud contains S(0, r), then S(0, oc) + Ε is contained
Proof. For any у e S(0, a) there exists а у' е S(Q, r) with у = (<x/r)y'. For any
y"eE
with y0e Ε. α
If dhEis the set of points in Rd having distance ^ h from the boundary of E, then
dhEnEc is contained in (E + 5(0, h))\E. For convex E, E\dhEis convex and it
contains a sphere of radius ρ (Ε) — 2h. Combining these observations with Day's
lemma it is very simple to prove the following auxiliary result of Nguyen-Zessin
[1976] and Fritz [1970]: If An is a sequence of convex sets in Rd with ρ(Αη) -* oo,
then λ (dh Αη)/λ (An) tends to 0 for all h > 0, and λ (Α™ \Α™)/λ (Αη) tends to 0 for all
meN.
A sequence of convex sets An will be called regular if there exists an increasing
sequence of intervals Γη in R+d and a constant С < oo such that Tn contains An and
λ(Γη) й СХ{Ап) holds for all n.
Theorem 2.12. If τ1?..., τά are commuting endomorphisms of (Ω, s&\ μ) and if
Au A2,... is a regular family of convex sets in R+d with ρ(Αη) -> oo, then
card(^nV)-1 Σ /°τΜ
ueAnη V
converges a.e. for all fe Lp, (1 ^ p < oo). Here AnnV may be replaced by any
other subset of У differing from AnnV only on dhAnfor some h > 0.
This is just another special case of the results of TempePman and the same proof
applies. Note that λ(δΗΑη)/λ(Αη) -+ 0 implies
cardan V)-1[card((ylnnV)A((Ann V) + ej)J] -+ 0 and the corresponding
result for sets differing from AnnV only on dhAn. Examples of such sets "close"
to AnnV are^nV and^nV.
Let Zwm be the set of vectors и e V(m) with [w, и + me[ c: An and let YHt m be the
set of vectors и е V(m) with [w, и + те\_слАп^ 0. Then we have
A™ = (J [u,u + me\_ and A™ = (J [w, w + me [.
Applying the theorem to the sub-semigroup {τΜ: ue V(m)} we find that
(2.11) Я(Л)"1 Σ /nH
converges to 0 a.e. for all integrable/
α
r + α
/+
r + α
r + α
■л

210 Multiparameter and amenable semigroups
It is now simple to prove the following ergodic theorem for additive processes
with a continuous parameter due to Nguyen and Zessin [1979]:
Theorem 2.13. Let τ = {τΜ: и е R+d} be a measurable measure preserving semi-
flow in (Ω, s/9 μ) and let F: @b э A -► FA e Lx satisfy (2.8) and (2.9) with TJ
= /°V Assume that
(2.12) there exists anfe Lx such that \FA\ rg/holds for all convex sets A e 38b
with A a [0, e [.
Then X{Ar)~1 FAn converges a.e.for any regular sequence (An) of convex sets with
q{An)-+co.
Proof By the assumptions \FAn — Ση€Χη ^ο,^0^! *s bounded by
HueYnA\Xn,J°tu· О
The assertion of theorem 2.13 remains true if the sequence (An) is replaced by a
family (Ar) indexed by any countable linearly ordered set, but again one must add
further conditions if r shall range through R+.
Let Жу denote the family of non empty finite subsets of V. A subadditive
process for {τΜ: и в V} defined on Жу is a set function F: Жу 3A-+FAeLx with
the properties
(2.13) FA о xu = FA+U whenever A e Жу and и е V;
(2.14) if Au A2 are disjoint sets in Жv then FMkjM rg FAl + FA2;
(2.15) y(F) = inf {card(^)"1 JFAdp. A e /} > - oo.
Note that the spatial constant is again defined with intervals!
Theorem 2.14. Let τ1?..., τά be commuting endomorphisms oj (Ω, s&\ μ) and let
F= {FA: А в Жу} be a subadditive process for τ = {τΜ: и е V}. If(An) is a regular
sequence of convex sets inR+d with ρ(Αη) -> oo, then card (АппУ)~х FAnnV
converges а.е..
The proof is virtually identical with that of theorem 2.9. The role of Ir m is now
played by A™ and the role of Jr m is played by A™.
In the continuous parameter version of this theorem, Fmust be defined on 3Sb
and (2.12) must be satisfied. Then one gets the convergence a.e. of X{Ar)~1 FAn as
above.
Notes
The idea of regularity of one sequence of sets with respect to another seems to appear first
in the paper of Pitt [1942], and was most successfully extended by Tempel'man. The proof
of lemma 2.5 uses an idea of Calderon [1953].

Semigroups of Lx-contractions 211
Also Fava and Nanclares [1978] and Becker [1981] have ergodic theorems for convex
sets.
The ergodic theorem for additive processes in Ud not given by integral averages is of
interest for the theory of point processes; see Nguyen-Zessin [1976], Rolski [1981a].
The question of a.e.-convergence for multiparameter subadditive processes was first
treated by Smythe [1976]. Smythe considered 2-subadditive processes which are also
strongly subadditive: For d = 2 this means that s = (sx, s2) < t = (tu t2) < и implies
F[S,u[ — %ι,ί2),«[ ~~ F[(tLS2)M + F[t,u[ й F[S,t[ -
Using this condition, Smythe proved an analogue of Kingman's decomposition theorem.
He obtained a.e.-convergence of n(n)~1F[0M for sequences remaining in a sector and
some first results on unrestricted convergence.
Nguyen [1979] studied groups τ = {τ,: t e Ud} and subadditive processes {FA} with A
ranging through the class of bounded Borel sets. {FA} is strongly subadditive in his sense if
fa^a2 + FAmA2= fa, + fa2 holds for a11 Ai> A2 with A1nA2=¥ 0. He obtained a
Kingman decomposition of processes with his property by a very different argument. He also
proved a.e.-convergence for sequences (An) of convex sets with ρ(Αη) -> oo, assuming his
strong subadditivity.
Krengel and Руке [1985] relaxed the convexity condition in theorem 2.14 and showed
that the convergence to the limit/of the sequence F[0ne[ is uniform in the following sense
sup {\n~dFnB-X(B)f\: Вe@}->0 a.e.
when J1 is a family of sets in [0, e[ with "small boundaries".
The cluster process of Grimmett is not strongly subadditive in either sense. Akcoglu and
Krengel [1981] have given an example of a subadditive process {Fj\ I e /} for d — 2 which
does not admit a Kingman decompsition.
Grillenberger and Krengel [1982] showed: If g is a real valued translation invariant
superadditive set function on the system of bounded Borel sets in Ud and
inf {g(B): В cz [0, e [} > — oo, then g(Kn)/X(Kn) tends to the same γ for all sequences Kn of
convex sets with ρ (Kn) -» oo. Thus, the limit property defining the spatial constant remains
true for convex sets. For strongly superadditive g this was shown by Robinson-Ruelle
[1967] and Nguyen [1979].
Applications of the multiparameter subadditive ergodic theorem to Mathematical
Physics have been given by Kirsch and Martinelli [1982] and by van Enter and van
Hemmen [1983].
§ 6.3 Multiparameter semigroups of Lj -contractions
We return to the study of ergodic theorems for multiparameter semigroups of
operators. We use the notation from §6.1. Let {Tu:ueV} be a semigroup of
Li — L^-contractions. We shall show that ^„/converges a.e. for all integrable/if
η tends to infinity in a sector of V. Dunford-Schwartz [1956] proved the
continuous parameter variant of this theorem by a method which they called
"circuitous". We follow the much simpler direct approach to the discrete case given by
Brunei [1973] and obtain the continuous parameter result as a corollary.
We also show by example that there is no such multiparameter generalization
of the Chacon-Ornstein theorem.

212 Multiparameter and amenable semigroups
Finally we sketch a multiparameter generalization of the stochastic ergodic
theorem asserting unrestricted stochastic convergence of Anfforfe Lx when the
Tu are positive Lx-contractions which need not contract the L^-norm.
Lemma3.1 For ξ(χ) = 1 — j/l — χ the coefficients a£° in the expansion \_ξ(χ)~]η
00
= Σ οί(ρ)χρ are given by
P = o
(3.1) af
"p
0 for ρ < η
n2n+^2pi2p-n-l\
2p \ p-\ )
00 /l/2\
Proof. The case η = 1 follows directly from (1 — x)1/2 = Σ ( — *Υ· It is
p=o \ Ρ /
then obvious that a£° — 0, {ρ < ή), holds for all n. Assume that the assertion of
the lemma has been proved for all η ^ N. The trick in the induction step (due to
N. Neumann) is to use the recursion
(3.2) ίξ(χ)Υ+1 = 21ξ(χ)Υ-χίξ(χ)1Ν-\ (Λ^Ι)
which follows from (ξ (χ))2 = 2 ξ (χ) - χ. For Ν = 1 and/? ^ 2, (3.2) yields α<>2) =
2ο4υ which easily implies (3.1) for n = 2. For TV > 1 and /? ^ TV + 1 we obtain
2* + 2-2p/ 2p-N-2
<+1) = 2<> - a^1»
N+^N+2-2p(2p-N-2\
2p " \ p-! )
ρ Ν-ί p-\
p-\ N+\ p-N
N-\
2(P-1)
TV
ЛГ+1
P-2
2p-N-\
p-N
Because of [...] =1 the induction is complete. □
Lemma 3.2. For any K^z\ there exists a constant c(K)>0 such that
a*f° ^ c(K)np~312 holds for all n,p with n2 ^pK.
Proof. Let us use a„ ~ b„ as a shorthand for
lim sup a„/b„ < oo and lim inf ajb„ > 0.
By Stirlings formula k\ ~ е~ккк+1/2. Because of (3.1)
«w~ n-1-2" &P-n-W n_ φ p)

Semigroups of L1 -contractions 213
n \ 2p-n-l Д n\-P + "
with c(n,p) = ( 1 — — 1 II J . Now
η η2 \ / η η2
1оёс(п,Р)~(2р-п-1)^---—2-...)-(р-п)^-р2р2
η2 η2 η3 \ ( η2 η2 η3
If η and/? tend to infinity subject to η2 ^ pKihe last expression remains bounded.
Hence c(n,p) is bounded below by a positive constant, α
Lemma 3.3. Let φ (ή) be the integer part ofyn + 1. There exists a constant c' > 0
such that
1 c'
holds for 0 S v, w < n.
Proof. Because of аЦЦ = 1 the term withy = 0 is large enough in the case ν = w
= 0. Iff = 0andw> 0, the term withy = 1 is ^ φ(η)~χ c{\) (w + l)"3/2? and this
is ^ c'n~2 for small enough c'. By symmetry it is now sufficient to consider the
r-l
case 0 < ν rg w < n. Using lemma 3.2 again and Σ Α:2 ~ r3 we then find
ι
φ{ή)-' Σ №j^+j^ Σ const Λ/ /^3/2
]<φ(η) j=i ((V + j) (W +7))3/2
^ φ{η)~ι const't;"3φ(V)3 ^ const" и-2, а
For d > 1 and η β Ν let Hd be the integer φτη(ή) with 2m~1<d<.2m. The basic step
in the argument of Brunei is:
Theorem3.4. For any d>\ there exists a constant xd>0 and a family
{a(u)\ ueV} of strictly positive numbers summing to 1 such that the following
holds: IfTu T2,..., Td are commuting contractions ofL1 and\ Tt\ denotes the linear
modulus of Th then the operator U = ΣΜε ν a(u) I T\ Γ1 · · · I Td \Ud satisfies for all
n^l andfe L\
nd-l
(3.3) n~d Σ ··· Σ \iil...rd*\fu- Σ ujf.
0^ίί<η 0^id<n nd j = 0
Proof We may assume d = 2m, because we can put 7J = /, (d+ί uju 2m) if
d< 2W. We start with the case T{ = | T{\ ^ 0.

214 Multiparameter and amenable semigroups
Put ξ(χ) = χ'1 ξ(χ) = Σ <4+1 *ν· For a contraction Γ, ξ (Τ) is defined by ξ (Γ)
ν = 0
= Σ α^λ Τ\ Clearly (ξ(T))j = Σ α#ν Γν. For /n = 1 (J = 2) we may take
v=0 v=0
υ=ξ(Τ1)-ξ(Τ2).
Then
φ{η)-χ Σ U^cpiny1 Σ Σ αψ+1ια%2ψΤ^
j<q>{n) 0^ii,i2<n j<q>{n)
^ Дг Σ 71'1 Ti\
ft 0^ii,i2<n
i.e., (3.3) holds with χ2 = с'. For m = 2,(d = 4), four contractions Tu ..., T4 are
given and we put
and
Then
υ = ξ(υι)-ζ(υ2).
Σ uJ^^-T2 Σ и\1и\г
ψ{φ{η)) ]<П4 φ(η)2 kl<„2
к2<П2
(с')3
ft jl<n J4<n
and we may take χ3 = χ4 = (с')3· In general, if we know the construction of [/for
d operators and Tu ..., T2d are given, let U1 be the operator constructed for
7j, ...,Td, let C/2 be the operator constructed for Td+l9...9T2d and take U
= f(t/1)-f(t/2).
[/is of the form [/ = ΣΜ«= ν я (и) 7jMl... Tdd with coefficients a(u) > 0 summing
to 1 which do not depend on Tu ..., Td. Let aU) (il9..., id) be the coefficient of
T{x ... Tdid in Uj. The inequality proved is due to the fact that
Xdn^ Σ a^(iu...Jd)^n-d
j<nd
holds for all (/1?...,/d). The numbers aU)(il9..., id) are sums of coefficients
of different products Ta{1) - Тл{2) ·... - Ta{s) coming from the evaluation of
Uj = (ΣΦ) TxUi... TdUd)j, where a(v) = 1 holds for ix values of ν, α(ν) = 2 for i2
values of v, etc. In other words: aU) (il9..., id) = Σαβ0,α) (h> ···> *d) where
α0,α) (/1?..., /d) is the positive coefficient of Тл(1)... 7^(s). The inequality (3.3)
remains true for non positive operators and / ^ 0 even though the operators
| 7j |,..., | Td | may not commute, because of

Semigroups of Lx-contractions 215
хл1 Σ u^junj1 Σ Σ Σ«0'·α4ίι,···Λ)|Γα(1)|...|Γα(5)|/·
0^j<rid j<rid ii,...,id<n л
^ Χ*η? Σ Σ Σ я(Ля) Οι, · · ·, id) Ι Τα(1) ·...· Tm \f
j<nd И» ...,id<n α
= Zdnj' Σ Σ «υ) (/ι, · ··, ϋ Ι ϊ?1 · · ·· · Tt\f
j<nd ii,...,id<n
^«-" Σ |γ/'·...·γ;<ί/. d
ίι, ...,id<n
It is now easy to derive the following multiparameter ergodic theorem for
Lx — L oo -contractions:
Theorem3.5. If Tx,..., Td are commuting Lx — L^-contractions, the averages
Au{k)f converge a.e.for allfe Lx and all sequences w(l),..., u(k),... tending to oo
in a sector ofV, in particular for u{k) = ke = (£, k,..., k).
Proof. We have to show that
МЛ = sup {\Aumf- AM(t)/|: / ^ A:}
tends to 0 for all/e L\. By theorem 1.2 Лк(/') tends to 0 for all/' e L2. Given
ε > 0 find f e L\ with /' £f and ||/-/'|li < ε2· Clearly
J*CO==limsup^004(^^sup^t(/-/')^2supJAuW(/-/')|. In the
case u(k) — к · e we may apply (3.3) directly and obtain
m-l
μ(Α4η>^^μ^ρχά2ηι-1 Σ υι\/-/'\>ε).
m i = 0
As U is a positive Lt — Z^-contraction Lemma 1.6.1 now implies
μ(Λ*(/)>ε) й 2Xds-1\\f-f'\\1<2Xde.
As ε > 0 was arbitrary this completes the proof for the case u{k) = ke. In the
general case put n{k) = max {ux (k),..., ud{k)}.
As the sequence и (к) is in a sector of V there exists a constant 0 < c0 < oo with
n(kyd й c0n(u(k)) for all k.
By the previous theorem
\Аи(к){/-Г)\йп(.и{к)у1 Σ ·.· Σ \Tl\..TdHf-f')\
ii<ui(k) id<ud(k)
Sc0n{k)-d Σ ... Σ Ι τ?... #-(/-/') I
i\<n{k) id<n(k)
ucoXd(n(k))^ Σ U>{f-f).
j<(n(k))d
The same argument as above now gives μ(ζ!*(/) > ε) ^2xdc0s. π

216 Multiparameter and amenable semigroups
A similar proof shows that card(2?knV) x Σ Tuf converges a.e. for all
ueBknV
fe Lu when Bk\s a sequence of convex sets with ρ (2?fc) -> oo which is regular with
respect to some increasing sequence of squares Гк = [0, n{k)e\_. We leave this as
an exercise. One first has to extend theorem 1.2 to averages over convex sets. In
the extension of theorem 1.2 regularity is permitted with respect to any sequence
I'k = [0> u(k) [ with u(k) -* oo in V, which need not be increasing. The results of
the previous section suggest that also for general Lx — L^-contrations Tl9..., Td
the averages ylM(k)/converge a.e. for all/e Lx and all increasing sequences u(k) in
V. This, however, cannot be proved with the method of majorization by £/and is
unknown.
What can be said about the identification of the limit in theorem 3.5? We assume
that the T{ are positive: Let Cv be the strongly conservative part of the positive
n-l
contraction U. As η -1 Σ E/'/converges to 0 for all/e Lx on the complement Dv
i = 0
of Q/ the averages Au{k)f must also tend to 0 on Dv. There exists an element/? e L\
with Up = p and {p > 0} = Cv. Lemma 2.1.14 of Brunel-Falkowitz implies that
p is invariant under all T{.
The next lemma shows that there is no influence of the dissipative part Dv:
Lemma 3.6. If С and D are the conservative and dissipative parts of a positive
Lx — L^-contraction T, then T\D ^ 1D.
Proof We know T\c = lc. Therefore 71^1 implies T\D л 1D = 0. α
As there is also no influence of Cv \ Cv in Cv the limit behaviour of the averages
depends only on the values of/in Cv and we may and do assume Ω = £υ. Now
the usual arguments show that the averages ylM/are uniformly integrable and one
has also Lx-convergence. The limit/must be invariant under all Tu, (и е V), and
hence ^-measurable. Jv is the σ-algebra of sets invariant under all TJ./is
uniquely determined by \^άμ = \ΈΥάμ for all Fe Jv.
We only sketch the extension to continuous parameter semigroups: If
{Tu: ue U+d} is a semigroup of contractions in Lx which is continuous in the
interior of U+d (i.e., || TJ- TJW^ -► 0 holds for и -► υ > 0 for all/e Lx) we may
define suitable representatives of Tuf (u > 0), as in the 1-parameter case (§1.6):
Put
fn(t,co)= T{2-nil9m..92-n^f((o)
for t = (tu ..., td) with tv e [2"w/v, 2~n(iv + 1)[.
The limit/(/, ω) of an appropriate subsequence defines representatives/^,.) of
TJ. Put
S0t0f= ί TJdu, A0tJ=n(v)~1S0tJ.
[0,v[

Semigroups of L1 -contractions 217
Theorem 3.7 (Dunford-Schwartz). If {Tu: и e R+d} is a semigroup of Lx — L^-
contr act ions, continuous in the interior ofR+d, the averages A0aef converge a.e.for
all fe Lx as a -> oo.
Proof An application of theorem 3.5 to the function S0ef shows that there is
a.e.-convergence when a takes only integer values. It remains to show
(3.4) lim (A0taef- A0t[a]ef) = 0 a.e..
Forfe L1nLa0 this is simple. The general case follows by approximation with
bounded functions once we show
(3.5) J lim sup \Α0^\άμ й XdWfWi ·
Recall that ej was y-th unit vector in V. Put
Λ(/,ω) = |Γ2-„β1|ίι·|^2-^Ιί2...|^2-^ΙίίίΙ/Ι(ω)
for / = {tu ..., td) with tv e [2-n/v,..., 2~n(iv + 1)[. This sequence is increasing
and L^bounded. If /is the limit and
F= J Ku.)dt
[0,e[
then
\S0,aef\u Σ ..· Σ \T{iu_id)\F,
and theorem 3.4 yields
lim sup \a~dS^aef\ й Xd lim sup η'1 "ς U*F.
a-* oo n-> oo i = 0
(3.5) now is a consequence of \\F\\l rg Ц/Ц^ п
The following example, obtained jointly with A. Brunei, shows that the Chacon-
Ornstein theorem does not extend to multiparameter semigroups.
Let d = 2 and let Tt for i = 1, 2 be given by a translation in Ω = Ζ (with
counting measure), i.e., Τ^{ώ) =f(co + 1). S^/has the form
5Μ/(ω) =/(ω) + 2/(ω + 1) + ... + /ι/(ω + и - 1) + (и - 1)/(ω + и) + ...
+/(ω + 2/ι-2).
We sketch the construction of non negative integrable functions /, g with
g(0) > 0, for which Snef/Sneg diverges in ω = 0: Let hk be the function which
equals l/]/k on {k — 1} and 0 everywhere else. For η ^ k, Snehk(0) = ]/k, and
for η < k/2, Snehk(0) = 0. Let (mj)j^1 and (nJ)j^i be two sequences of integers
00 00
withw1 = l,«J>2mJ,m;+1>2«J+1,0^1).Then/= Σ A»>andg= Σ Amj.are
integrable, and ■/_1 J~1

218 Multiparameter and amenable semigroups
Smjg(0) = Snj.g(0) =
smjf(0) = Σ1 ]/щ,
s„J(0) = Σν4.
: Σ
k = l
k?
k = l
The sequences may therefore be chosen in such a way that the lim inf of Snef/Sneg
at 0 is 0 and the lim sup is oo.
With some more effort one can construct functions /, g for which the same
ratios diverge everywhere. Using stacking constructions (see Friedman [1970]) it
is possible to find a conservative Tx = T2 and functions/, g for which the ratios
diverge everywhere.
Using the same operators as above it is also easy to construct integrable and
strictly positive functions / g with
Σ 71М1Г2М2/=оо and Σ ?Γ Tpg < oo
«eV ueV
on Ω = Ζ. Thus, there is no Hopf decomposition.
Proposition3.8. Let Tu ...,Td be commuting positive contractions in Lv If Q
d
denotes the conservative part of Ω for Th then Cd = f] Q is Tj-absorbing for j
= 1,...,</.
Proof We may assume d = 2. If C2 is not 7i-absorbing there exists fe L\ with
{/> 0} с С2 and a set A a Cc2 η Q of positive measure with Txf> 0 on A. As
Σ 72'/ diverges on {/> 0} we see that
i = 0
ΙΙ((Σγ2'/)λα:λ-*/ΙΙι-ο, (jv^<x>)
i = 0
for all AT > 0. Hence, on A
т1(1.ПЛ=1.П(т1л-*оо, (tf-oo).
i = 0 i = 0
oo
On the other hand TJeLx and Л с C$ imply Σ Ti(Txf) < oo on yl, a
contradiction, α ί = 0
For the multiparameter generalization of the stochastic ergodic theorem we need
a multiparameter generalization of Neveu's decomposition of Ω into the strongly
conservative part С and its complement:
If Tl9 T2,..., Td are commuting positive contractions in Lx a function h e Lj is
called weakly wandering for {Tu:ue V} if there exists a sequence w(l) < u(2) < ...

Semigroups of Lx-contractions 219
00
in V with 11 X T^k) h \ \ ^ < oo. h is called m-weakly wandering if there exists such a
fc=l
sequence having the stronger property
sup||EM(k)^i;^(ek)-^||00<oo.
reV
Theorem 3.9. For the semigroup {Tu:ueV} there exists a decomposition of Ω into
disjoint sets C, D which is determined uniquely mod μ by the properties:
(i)' There exists p0 e L\ with {p0 > 0} = С and Ttp0 — p0 for i = 1,..., d;
(ii) There exists an m-weakly wandering h0 e L* with {h0 > 0} = D.
We do not give the proof though it contains some technical difficulties, because in
essence it is very similar to the 1-parameter special case treated in § 3.4. The role
of the conservative part of a single operator is now played by Cd.
Theorem 3.10. Let Tl9 T2,..., Tdbe commuting L ^contractions. Assume that also
the linear moduli \ Tx |, | T21,..., | Td\ commute. Then, for fe L1? the averages Auf
converge stochastically as и -> oo.
Sketch of proof We apply the previous theorem to the semigroup generated by
| Tx |,..., I Td |. Let £"and Fdenote the sets with Ε = {p0 > 0} = Fc. Eis absorbing
for the contractions \T(\ and hence also for 7]. For/e Lx with support in Ε the
existence of p0 e Lx with | Ttp0 | ^ p0 implies even convergence in the Z^-norm.
To prove stochastic convergence to 0 in Fwe may assume 7J ^ 0 because of | Tuf\
^ \Tx\Ul... \Td\Ud\f\. As in the 1-parameter case one can then show that
II A*h 11да -> 0 for weakly wandering h and this again yields the desired stochastic
convergence on F. The only new difficulty is to get control of the influence of Fin
E. Put U = Tx · T2 ... Td. Forfe Lx the functions dk = dk(f) and rk = rk(f) are
defined inductively by d0 = iEfr0 = iFfdk+1 = lEUrk,rk+1 = lFUrk. The maps
η
f-+Knft= Σ db in = oo, 0,1, 2,...) are contractions mapping ΕΧ(Ω) into
i = 0
LX(E). It suffices to prove || lEAu(f- K™/)^ -^ 0 for u-+oo.
We write/ ~>x f if there exists a measurable g with 0 rg g rg 1 and an integer i
with 1 ^ / ^ J and/' = gf+ 7]((1 — g)f).f~>/' shall mean that for some η there
are functions/1?/2, ...,/„ with/-»! ft ~>x f2 ~>... ~>x /и = /'. By a cancellation
argument one easily sees that/^/' implies Ми(/—/')lli ~* 0. By induction
this remains true for/~»/'. One can check that f~>(KNf+ rN) holds for all N.
Because of \\KNf— Kcof\\1 -^ 0 it is therefore enough to show that || Ι^^ΙΙ!
tends to 0 uniformly in и for N -* oo.
The sequence ll^lli decreases to some η ^ 0. We show that Hr^Hi — Ц <£
implies Ρ^Τ^Ι^ ^ ε for all veV. Pick someL ^ Max {t;1? t>2?..., fd}. As J?is
7]-absorbing for all i we have
rN + L—iFU rN~iF1l 22 -^d UfA ··· ιά rW'

220 Multiparameter and amenable semigroups
Hence η й ||rN+L||1 й Ρ^^νΙΙι = II^^IU - \\1εΤόγν\\χ й \Ы\,
- WUT^W, and WUT^W, й \\τΝ\\χ-η. Π
Notes
Dunford and Schwartz have used a construction of continuous time 1-parameter
semigroups from 2-parameter semigroups which has inspired the construction of (/in theorem
3.4. They do not give an inequality between operators as in (3.3). But in principle, their
construction could also be used to prove theorem 3.5 and it remains useful for local
ergodic theorems.
The assertion of theorem 3.5 holds also for positive Lx-contractions Tu ...,Td with
|| Ti\\p 5Ξ1 for some ρ > 1. This was shown for d = 1 by Akcoglu-Chacon [1965] and the
general case was reduced to the case d = 1 by McGrath [1980a]; see also Sato [1983].
Combining theorem 3.4 with theorem 1.6.3 we obtain the following theorem which goes
back to Wiener in the special case of measure preserving transformations.
Theorem 3.11. Let {Tu: и e V} be a semigroup of Lx — L^-contractions in Lx of a finite
measure space. For поп negative fe LlogL,/* = sup{«_dXM<ne| Tu\f} is integrable.
Extending Ornstein's result of §1.6., K.Petersen [personal communication, 1977] and
Nghiem Dang Ngoc [1980] have proved a converse for this theorem for ergodic
^-parameter groups of measure preserving transformations: For non negative / the in-
tegrability of/* implies fe LlogL.
Derriennic and Krengel [1981] proved the multiparameter Li-mean ergodic theorem
for superadditive processes:
Theorem 3.12. Let {Tu,ueV}bea d-parameter semigroup of positive L ^contractions with
Ω = С and let Fj= {/}: Ie /} cz Lx satisfy TUFM = Fl + u, (Ie f,ue V); (2.2) (with n = 2);
and (2.3), then F[0n[ = π (и)-1 F[0n[ converges in L^norm to an invariant limit ffor n -» oo.
Akcoglu and Sucheston [1983] proved a multiparameter superadditive stochastic ergodic
theorem: If {Tu: и e V} is a ^-parameter semigroup of Markovian operators and F is as
above, then F[0n[ converges stochastically. The limit is 0 in Δ It is easy to deduce this from
theorems 3.10 and 3.12 (to whichthey had access), but they have a direct argument which
replaces the characterization of С in theorem 3.9 by a different characterization for
general families of contractions which admits a shorter proof.
An application of theorem 3.12 can replace the argument for the convergence on С in
theorem 3.10. The original argument was prefered here, because it gives access to the
identification of the limit.
Emilion and Hachem [1983] have proved a.e. convergence of F[0 ke[ for Markovian Tu
when {Fj} is "strongly superadditive". They also discuss the submarkovian case and
processes in Lp, (1 <p < oo).
Finite invariant measures. Blum and Friedman [1967] and Hanson and Wright [1970]
started the search for conditions for the existence of finite invariant measures for Abelian
semigroups of nonsingular transformations, and Granirer [1971] gave an extension of the
first four conditions in Corollary 3.4.7 to the case of amenable semigroups of
transformations. Sachdeva [1971] and Takahashi [1971] proved for general left amenable
semigroups of positive contractions in Lx that there exists a strictly positive invariant/e Lt if
and only if there exists no weakly wandering ЛфО. Takahashi also constructed a strictly

Amenable semigroups 221
positive weakly wandering h for left amenable semigroups admitting no invariant fe L\
with/Φ 0. In the special case of semigroups {7^: и e V} their results follow from theorem
3.9. They do not give the decomposition Ω = Си D needed for the proof of theorem 3.10.
The omitted proof of theorem 3.9 along the lines of the 1-dimensional case uses tools of
their papers and has been carried out in detail by my students, Mrs. Kleinknecht and
Mr. Pelanchon.
Hiai and Sato [1977] have studied С and mean ergodic theorems for semigroups of
positive contractions in Lv
Hajian and Ito [1969] have extended the result of Hajian-Kakutani on weakly
wandering sets to general groups {ig\ge G} of nonsingular transformations: There exists an
invariant probability measure ν equivalent to the given measure μ if and only if there exists
no weakly wandering A with μ(Α) > 0. Here, A is called weakly wandering if there exists a
sequence gl9 g2,... in G for which the sets τ"1 A are disjoint. A related reference is Dang
Ngoc Ngiem [1973]. J.M.Rosenblatt [1974] derives sufficient conditions for the
existence of an invariant ν ~ μ for finitely generated groups.
M. Wolff [1971] has considered the existence of vector valued invariant measures for
amenable semigroups of operators.
J.M. Rosenblatt [1981] proved for some compact Abelian groups Xthat the group G of
topological automorphisms of X has Haar integral as unique G-invariant mean. (This has
been used by Sullivan [1981] and Margulis [1980] to show that for η > 3 Lebesgue
measure is the only finitely additive normalized measure on the «-sphere invariant under
rotations.)
Further references are given in the notes of §3.4 and in §4.3.
§ 6.4 Amenable semigroups
We now consider a class of semigroups of operators more general than the
d-parameter semigroups, and containing many non Abelian semigroups. This
class constitutes a natural general setting for mean ergodic theorems. We need
further restrictions for a pointwise ergodic theorem for measure preserving
transformations.
Throughout this section, G will be an abstract semigroup and 0 a σ-algebra of
subsets of G such that for и e G and В е У the sets Bu, Bu'1 ··= {v e G: vu e B) are
^-measurable. We assume the existence of a σ-finite measure ν on (G, <&) which is
right invariant (v(Bu) = v(jB)). E.g. G can be a locally compact group and ν the
right Haar measure.
1. The mean ergodic theorem. We first derive a mean ergodic theorem. Let X be a
(real or complex) Banach space, and let 5^ = {7^: и е G} be a bounded semigroup
of continuous linear operators in X. (We tacitly assume Tuv—Tu Tv). Put
M-=sup{\\Tu\\:ue G} <co. <Sf is called weakly measurable if the maps
и -* (Tux, hy from G into the scalar field of£ are measurable for all χ e 3E, he 3E*.
For В with v(B) < oo let S(B)x denote the element of £** with
(4.1) <S(B)x, h> = J <Гмх, A> v(du).
в

222 Multiparameter and amenable semigroups
If χ is an element of X for which the weak (= strong) closed convex hull
Cx-= со £fx is weakly compact, then S(B)x belongs to X с £**: To check this we
may assume ν (В) = 1. Cx, considered as a subset of £**, is compact in the
σ(£**, £*)-topology. If S(B)x does not belong to Cx, there exists an h e X* with
Re(S(B)x, h} > βχ:= sup {Re<>>, h}:yeCx},
(see e.g. theorem V.2.10 in Dunford-Schwartz [1958]). This contradicts (4.1).
For any measurable E<^G we have (BnEu~x)u = Bun Ε and hence
v(BnEu~1) = v(BunE). Because of the identity lEu-i(v) = lE(vu) this means
that
(4.2) if(vu)v{dv)=i№v(dv)
В Ви
holds for indicator functions/ = 1E. Thus (4.2) holds for all bounded measurable
/on G. Applying (4.2) with/(t;) = <ΤΌχ, h} we find
(4.3) S(B)Tux = S(Bu)x.
We shall consider sequences I = (/„) of measurable subsets of G having some or
all of the following properties:
(PI) 0<v(/„)<oo, (и = 1,2,...);
(P2) For all ue G, limnv(InAInu)/v(In) = 0;
(P3) /„^/п+1,(и = 1,2,...);
(P4) There exists Kx < oo with ]imnv*(INQ/v(Q й Кг for all N;
(P5) There exists K2 < oo with v*(I~xQ й K2v(In), (л = 1, 2,...).
(Here v* denotes the outer measure, and B^1 B2 the set {ve G: 3we B1 with
wt;e B2}.)
If (PI) holds, put A[x = v(/„)_1 S(In)x. If Cx is weakly compact for all x9 the
yl£ are linear operators in X with norm ^ M, and ^Jx belongs to Cx. For any χ
and и there exists an h e £* with ||h\\ = 1 and
= v(/n)-4 J <Γ,*,Α>ν(Α>)-J <Tvx,h}v(dv))
Inu In
= v(Iny1HInuAIn)2M\\x\\.
Applying Eberlein's theorem (Theorem 2.1.5) and the splitting theorem 2.1.9, we
obtain the following mean ergodic theorem:
Theorem 4.1. Let ν be right invariant and let £f be a weakly measurable bounded
semigroup in X. Let (/„) satisfy (PI) and(P2). IfCx = со Ух is weakly compact for
all x, the Alnform a right-У-ergodic net, and the strong limit lim Alnx — χ exists for
each χ. χ belongs to the fixed space F(£f) = {y e X: Tuy = у for all и e G}, and χ
— χ belongs to the closure of the linear subspace N = lin {(7^ — I)z: ue G,ze X}.

Amenable semigroups 223
Remarks. 1) The compactness condition is satisfied if X is reflexive. It is also
satisfied if X is the space Lx of a finite measure space and the Tu are L1 — L^-
contractions.
2) \\А{Тих — A^x\\ -> 0 implies Alny -> χ for any у = Тих, and hence for all
yeCx. In particular χ is the unique fixed point in Cx. If follows that if (Jn) is
another sequence of sets with the same properties, then lim Alnx — lim^x.
Let Cb(G) denote the space of bounded continuous functions on G.
The existence of sequences (/„) with the properties (PI) and (P2) has been
studied in the theory of topological groups. If G is a semitopological semigroup
and/belongs to the space Cb{G) we put/M(t;) =f(vu) and uf{v) =f(uv). A
positive functional ρ e Cb(G)* is called right (left) invariant mean if it satisfies <H, ρ>
= 1 and </M, ρ> = </, ρ> «M/, ρ> = </, ρ» for all и е G and all/. G is called right
amenable if a right invariant mean exists, and amenable if a (right am/left)
invariant mean exists. A concise introduction to the theory of amenable groups has
been given by Greenleaf [1969]. Abelian, and, more generally, solvable groups,
and compact groups are amenable. If G is a σ-compact locally compact right
amenable group (with right Haar measure) there exists an increasing sequence
(/„) of compact sets with union G which satisfies (PI) and (P2), and conversely the
existence of such a sequence implies the right amenability of G (see Emerson and
Greenleaf [1967]). If one allows more general avering procedures, more can be
said:
A right amenable 5^ (with the weak operator topology) always admits a right
^-ergodic net, c.f. Sato [1978], Day [1969]. However, we shall need averages
over sets In in the discussion of pointwise convergence.
2. The maximal ergodic theorem and pointwise convergence. A semigroup {τΜ}
= {τΜ: и e G} of endomorphisms τΜ of a σ-finite measure space (Ω, s/, μ) is called
measurable if the map (и, ω) -* τΜω is (0 ® s/, ^-measurable. Then the
semigroup У — {τΜ} defined in Lp, (1йр й оо), by Tuf = f°τΜ is measurable.
Let 3F be the family of sets Ie У with 0 < ν (I) < oo. A process F= {Ff. Ie^}is
called super additive for {τΜ} if each FI belongs to Ll9 InJ = $ implies
Fj + FjUF^j^Fj^Fjot», (иеС,/еП and
y(F) = sup {v(I) x \Fjdp.Ie ^} < oo. Fis called subadditive if — F'\s superad-
ditive, and additive if Fis sub- and superadditive. For any/e L1 the process Ff
= {F/:Ie^} with
F/{<D) = lf{Tv<D)<dv)
I
is additive with y{Ff) = J /άμ.
Theorem 4.2 (Maximal ergodic theorem). Let G be a semigroup with right
invariant measure v, and {τΜ: ueG) a measurable semigroup of endomorphisms of the
σ-finite measure space (Ω, sf, μ). Let (In) be a sequence of subsets ofG with (Pl)-
(P5), and let F be a non negative superadditive process. Put Fk = v(/k)_1ivk and

224 Multiparameter and amenable semigroups
£:={й)6 Ω: sup (Fk((o): 1 rg к rg N) > a}, wAere TV is аи arbitrary integer and
a > 0. ГАел /ι(Я) ^ KxK2y(F)/a.
The proof is similar to that of theorem 2.6. We begin with
Lemma 4.3. Let A be a measurable subset of G for which INA is contained in a
set В of finite measure. Let k\u -> k{u) be a map of A into {1,..., N}. Then there
exists a finite subset А с A for which the sets Du ·-= Ik(u) и are disjoint such that A is
contained in the union of the sets Du := I^)h{u)u^ (u e A')-
The proof of this lemma is identical to that of lemma 2.5. The collections Ji-} are
finite because of ν (Du) ^ ν (Ij) and Du с В.
Proof of theorem 4.2. First fix some ω and η and put A = Α (ω) :=
{ueln: τηω e E}. For any и e A there exists k(u) e {1,..., N} with Fk(u) (τΜω) > α.
Let В be a set of minimal measure containing INIn. If Μ (ω) is the union of the sets
Z>M, (we yl')? obtained from lemma 4.3, property (P5) yields ν (Α (ω))
й v*(\JueA'DJ = Κ2ν(Μ(ω)). As the sets Du are contained in B, the properties
of F imply
ЯИ^ Σ ^Μ(ω) = Σν(Α)^(Μ)(τΜω)^αν(Μ(ω))?
and hence ν (Α (ω)) ^ α _1 K2 FB (ω). If we integrate the left side and use Fubini we
obtain
\ν(Α{ω))μ(άω) = J μ {ω: τΗωβ Ε} v{du) = ν(Ιη)μ(Ε).
In
On the other hand we have \ΡΒ{ω)μ{άω) <; y(F)v(B) = y(F)v*(INIn). Hence
μ(Ε) ^ a"1 K2y(F)v*(INIn)/v(In). The assertion now follows from (P4). α
Theorem 4.2 is due to TempePman for additive F. The proof of a pointwise
ergodic theorem for superadditive processes with general G is an open problem,
but the additive case is now easy:
Theorem 4.4 (Tempel'man). Let C?, {τΜ}, (/„) satisfy the assumptions of theorem
4.2, andfe Ερ(μ) for some ρ with 1 rg ρ < oo. Then A^f converges а.е..
Proof Using theorem 4.1 and 4.2 we can extend the proof of theorem 2.9 to the
present situation, π
It is clear that theorem 4.2 yields also a dominated ergodic theorem for additive
F. We only sketch the arguments. Put M(f) =/ ν sup„^/and с = Kx K2 + 1.
Then μ(Μ(β > ос) й col'1 J/+ holds for/^ 0, and hence for all/with/+ e Lx.
In particular, μ(Μ(/) > 2α) = μ(Μ(/— ос) > α) <; са-1 ]"(/— α)+· Replacing α

Amenable semigroups 225
by a/2 we obtain for/^ 0:
(4.4) μ{M(f)>u)^2c(χ-1 J /άμ й 2са~' J /άμ.
{2/>α} {2Μ(/)>α}
If we write X =/and У = Μ(/), the computation in the proof of lemma 1.6.2,
carried out with the new estimate (4.4), leads to
2Υ(ω)
(4.5) \ψ{Υ)άμ^2€\{Χ{ω) J rx\p(dt)}μ(άω).
о
Now the argument in the beginning of the proof of theorem 1.6.2 implies
(4.6) ||Г||р^2*-2-^||/||р, (ί£ρ«χ>).
For the I^-norm we obtain the estimate
00 00
|| Y\\1 й μ(Ω) + J μ(Υ> oc)doc й /i(fl) + 2с J α-1 J /(ω)άμάα
1 1 {2/>α}
^/i(fl) + 2cj/ / α_1ώφ = /*(&) + 2cJ/log+(2/)^.
ι
Notes
Day [1942] proved mean ergodic theorems for Abelian semigroups. Calderon [1953],
Tempel'man [1962-1974], Greenleaf [1973], Chatard [1970], Bewley [1971], and others
have studied mean ergodic theorems for amenable and more general semigroups, some of
them unaware of the work of Eberlein. Implications between measurability and continuity
properties of semigroups are discussed in the memoir of Moore [1968].
Nagel [1973] used theorem 2.1.11 to deduce the existence of an ^-absorbing projection
in со У for amenable bounded £f on a reflexive Banach space from the existence of fixed
points (due to Day [1961]). This does not involve the construction of ^-ergodic nets -
which may be an advantage in some respects. Along similar lines, Nagel-Palm-Derndinger
[1984] proved:
Theorem 4.5. If Sf is bounded right amenable, then the following assertions are equivalent:
(a) со £f contains an У-absorbing projection,
(b) co^x contains a fixed point for each x,
(c) The fixed points of У separate the fixed points of £f*.
Related results appear in Takahashi [1972], and Kijima-Takahashi [1973].
Amenable semigroups play an important role in the Jacobs-Deleeuw-Glicksberg
theory. It was shown by Deleeuw-Glicksberg that the sets Xrec and Xfl are ^-invariant
subspaces iff Sf is amenable. An application to Banach lattices appears in Schaefer [1974].
Itoh, Korezlioglu and Takahashi [1973] have characterized extremal invariant means
for semigroups Sf on the space of bounded measurable functions by ergodicity.
The problem of constructing avering sequences (/„) and averaging measures (weighted
averages) has been discussed by Oseledee [1965] and Tempel'man [1967].
Blum and Eisenberg [1974] call a sequence (ρη) of probability measures on a locally
compact Abelian group G generalized summing if, for every group {Tu,ue G} of unitary
operators Г„опа Hubert space § and every χ e § the averages ]Tuxon(du) converge

226 Multiparameter and amenable semigroups
strongly to an invariant element of §. They show that this property is equivalent to each of
the following conditions
(i) For every character χ on G with χ φ 1 the Fourier transforms ρη(χ) converge to 0;
(ii) ρη considered as restrictions of measures on the Bohr compactification G of G
converge weakly to Haar measure on G.
If (/„) satisfies (PI) and ρ^(Β) = v(BnIn)/v(In) then (P2) clearly implies that (ρ„0 is
generalized summing.
But one can construct sequences kx < k2 < ... so that /„ = {kl9 ...,kn} does not satisfy
(P2), but (ρ*) is generalized summing, see Blum-Eisenberg-Hahn [1973], Blum-Cogburn
[1975]. For related results, see Blum-Reich [1976], [1982].
The class of groups, for which sequences (/„) with the properties (P1)-(P5) exist, is fairly
small. (If G is countable, essentially the only such groups are those with polynomial
growth, i.e., finitely generated groups for which the size of the sets Wn of elements repre-
sentable by a product of at most η generators is bounded by a fixed power of n. Then one
can take In= Wn.)
On the other hand, (PI) is essential for the definition of the averages, and examples of
Tempel'man [1972] show that (P2), (P3) cannot even be omitted for G = Z, and (P5) not
for G = Z2. (For G = Zd, (P4) is a consequence of (P2)). In these examples/is bounded
and μ a probability measure. Greenleaf [1973] has shown that without (P5) a.e.-
convergence of A*nf may fail even for G = Z, using an unbounded integrable/. The
following theorem provides many examples having both properties, G = Ζ and \f\ <Ξ 1. In
addition, it sharpens an example of Krengel [1971], showing that the individual ergodic
theorem for subsequences fails for all aperiodic automorphisms; see § 8.2.
Theorem 4.6. There exists α sequence (/„) of subsets of 2. with (P1)-(P4) such that for all η
the elements ofIn+l\I„ are all larger than the elements ofIn, and such that for all aperiodic
automorphisms τ of a probability space there exists anf= lB>for which the sequence Alnf
(with xu = τ") fails to converge а.е..
Sketch of proof Ix := {0}. Step η + 1: Ih (i = 1,..., N„) are known. Write N = Nn. Choose
k larger than the absolute value of all elements of these Ih I = 2n + 2, and an extremely big
m. Rohlin's lemma yields a D (= D(n)) for which Д = τ* Д (i = 0,..., 2k(I + \)m - 1),
are disjoint and cover most of Ω. С = С (η) shall contain the first 2k sets Dh then it omits
2kl of them, then it contains the next 2k (second block), omits 2kl again, etc. Choose the
first к elements of IN+1 \IN consecutively so that τ7 carries the first half of the v-th block
into block ν + 1. Then choose the next k new elements^ so that the first half of block ν goes
into block ν + 2 (except for some blocks at the end). If IN+l is obtained in this way adding
sufficiently often k elements, then ^+11с(ш)1S cl°se t0 1 i°r ω m tne first halves of most
blocks. The next set IN + 2 is constructed similarly to make the averages large in the second
halves of most blocks. Then choose IN+3 so as to produce big averages on the first stretch
of length к of the gaps between blocks. After the construction of /Nn+1 with Nn+1
= J/V + 2(/+l)we know that on most of Ω at least one of the averages constructed in this
step is close to 1.
After completing all steps put В = и С (η). Then lim sup A[f = 1 a.e.. But \ίάμ < 1 and
the mean ergodic theorem shows that we cannot have a.e.-convergence. D
The condition on the In+1\In could be replaced by the condition Ζ = и /„, and one could
obtain divergence a.e. by minor modifications of the construction.
The proof above is similar to that of Krengel [1971]. While it was written, A. Bellow
[1984] showed that one could take In\In^ = {2n, 2n + 1,..., 2n + «}.
Although (P5) cannot be omitted in Tempelman's theorem it is not a necessary con-

Amenable semigroups 227
dition for a.e.-convergence. In fact, for some groups, Greenleaf and Emerson [1974] have
succeeded in constructing sequences (/„) for which a.e.-convergence of ^4^/holds for/e Ll
(and (P5) fails) by completely different techniques. They appeal to the non commutative
ergodic theorems of section 6.1 and show that for certain groups there exist sequences (/„)
for which Aln has the form of the averages discussed there.
Chatard [1970], [1972] and Bewley [1971] obtained results similar to Tempelman's
after the announcement of his main results but before his proofs were available. Further
related results were obtained by Emerson [1974], [1974a], and Tempel'man [1974].
The proof of theorem 4.2 uses arguments on G for fixed ω and then a passage to Ω via
Fubini. Calderon [1968] has shown that this idea (appearing already in Wiener's work)
admits a generalization to general principles, showing that maximal estimates in G
translate into more general estimates for measure preserving transformations.
The paper of Ornstein-Weiss [1983] contains a short proof of theorem 4.4 for countable
groups G.
Lin [1977] has considered operators of the form U = \GTgv{dg), where ν is a regular
probability measure on the Borel set of a locally compact Abelian semigroup G and
{Tg: ge G} a strongly continuous semigroup of contractions in a Banach space X. He
related the convergence of Unx to assumptions on the support of v.
Luczak [1974] has given a spectral representation of left stationary processes over a
separable locally compact group G of type I, and he has proved a mean ergodic theorem
when G is the group of linear transformations of the line.
Finally, we mention some interesting ergodic theorems for the group of rational rotations.
n-l
Let / be measurable on {z e C: \z\ = 1}, zn = e2ni/n and fn(z) = η'1 Σ f(zzn)· Jessen
fc = 0
[1934] proved the convergence a.e. of/Bi for fe LX(X) and sequences η = (щ) with
«i+1/«je N, called chains. Rudin [1964] snowed that/„ need not converge a.e. for fe L^\
more specifically, he showed that for any subsequence (pt) of the primes there exists an
fe L^ such that/p. diverges a.e.. Jessen's result has been improved by Baker [1976] and by
Dubins and Pitman [1979]. For K, L с N let Κ ν L = {к ν /: к е К, I е Ц where к ν lis
the smallest common multiple of к and /. The dimension d of К с N is the smallest integer
such that there are chains Ku ...,Kd with К с Κί ν ... ν Kd. Dubins and Pitman showed
that/„. converges a.e. for all/e L \ogd~1L if (щ) has dimension d. n
A related problem deals with the convergence of B„f(w) = n~l Σ/(^ίω)? where к
i = l
= (ki) is a strictly increasing sequence of integers and/is periodic on U with period 1. The
averages /^'converge a.e. for Riemann integrable/, but Marstrand [1970] showed that
B*f may diverge for bounded measurable / even in the special case kt = /.
Both Rudin's and Marstrand's examples may be considered as counterexamples to
ergodic theorems for Abelian semigroups with countably many generators.
Kieffer [1975a] and Emerson [1975/76] considered the problem of convergence of
A(v4„)-1 S(An) for strongly subadditive set functions S on unimodular amenable groups
with Haar measure λ.

Chapter 7: Local ergodic theorems and
differentiation
Let {Tt, t>0} be a semigroup of linear operators in Lp. Under suitable
assumptions
ь
Sa,bf=$Tsfds
a
is well defined for all/e Lp. Put A0a = t~l S0tt. Local ergodic theorems assert
that
(0.1) A0tf converges a.e. for all/e Lp as / -* 0
under various conditions, or they give an analogous conclusion for the
multiparameter case. They may be considered as results on the differentiability of
indefinite integrals.
We also look at a more general setting. Recall that a family F= {Ft, t > 0} of
elements of Lp is called an additive process for the semigroup if it satisfies: Ft+S
= Ft + TtFs, (t, s > 0). Clearly Ft = S0ttf is an example. If Ft = (Tt - I)f9 then
lim, _+0 + 0/-1i^-ifit exists - is the usual derivative of/, when { 7^} is the semigroup
of translations in R. In § 1 of this chapter we study the a.e.-convergence of t~l Ft as
/ -> 0 + 0 for additive processes and semigroups of positive operators. In § 2 we
discuss extensions to non positive operators, multiparameter semigroups and
functions / taking values in a Banach space.
§ 7.1 Positive 1-parameter semigroups
1. Two decompositions of Ω. Let У = {Tt, t > 0} be a locally bounded strongly
continuous semigroup of linear operators in Lp of a σ-finite measure space
(Ω, j/, μ), where 1 ^ ρ < oo is fixed. Recall from §1.6 that the integrals Satbf may
be defined either using suitable representatives/^ (/, ·) of Ttfor as the limit of the
Riemann sums. E.g., S0bf is the limit of
Μη)/=Φ/η)ΗΣΤίΐΗ/.
i = 0
We do not assume that the semigroup is strongly continuous at 0.
By a, process in Lp we mean a family F = {Ft91 > 0} of elements of Lp. F'\s called
positive if Fte L*, (t > 0), and strongly continuous if \\Ft — Fs\\p -> 0, (/ -> s > 0).

230 Local ergodic theorems
As the Ft are equivalence classes mod μ the question of a.e.-convergence of
t~l Ft makes sense only when / ranges through a countable set or when we refer to
specific representatives. A function F: ]0, oo[ χ Ω is a representative for Fif Fis
measurable with respect to the obvious product-a-algebra and if, for each / > 0,
F(t, ·) is a representative of Ft. It is easy to see that every strongly continuous
process has a representative. Note that an additive process F with \\Ft\\p -> 0,
(/ -> 0), is strongly continuous.
In this section we shall assume that the operators Tt are positive. A positive
additive process must then be non decreasing, i.e., Ft ^ Fs holds for t <s.It is easy
to see that a strongly continuous non decreasing process Fhas a non decreasing
representative F: F satisfies F(t, ω) ^ F(s, ω) whenever 0 < t ^s and ω e Ω.
If Q is a countable dense subset of R + \ {0} we say that Q-1imt^0ft exists a.e. if
the limit exists a.e. when / ranges through Q. If Fis a non decreasing
representative of F the assertions ^g-lim^of"1/J exists a.e.^> and <(limt_>0F(t, ω) exists
for a.e. ω^ are entirely equivalent. It will therefore be convenient to forget about
the representatives and to speak only about (^-limits. (Another way of dealing
with these technicalities is to use separability as in § 1.6 or essential limits; see the
Notes. For local ergodic theorems they do not appear since S0taf(co) is a
continuous function of a.) Before we can state the main result we must describe two
decompositions of Ω:
Let h e Lj with И > 0 a.e., and let h = S0tlh'. C0 ·-= {h > 0} is called the
initially conservative part ο/Ω, and D0 — {h = 0} is called the initially dissipativepart.
It follows from the following lemma, that this decomposition does not depend on
h:
Lemma 1.1. Ttf vanishes on D0for all fe Lp and all t>0. If C0 is contained in
{g > 0} for some g e L*, then S0tg is strictly positive on C0 for all t>0.
Proof We may assume/^ 0 and / ^ 1. The continuity of the semigroup implies
that Tth vanishes on D0. Now Ttf= lim„ Tt{f /\ nh). If the second assertion is
wrong there exists a subset A of C0 having positive measure, and an η > 0 with
Teg = 0 on A for 0<ε<η. For / > ε, Tt^eh vanishes in D0. Hence Tth
= T£ (lim„{ng a Tt_eh')) = 0 on A. As ε was arbitrarily small we obtain Tth = 0
in A for all / > 0, a contradiction to A a C0. α
We say that all mass in A disappears instantly if || S0ttf\\p is = 0 for all/e Uv (A)
and all / > 0. Clearly there is a maximal set D* mod μ in which all mass
disappears instantly. Put C* = Ω\Ζ>*. By the strong continuity of the semigroup,
fe LP(D*) implies Ttf=0 for all t > 0.
Obviously D* is empty if the enlarged semigroup {Tt, t ^ 0} is strongly
continuous at / = 0 and T0 is the identity. For other choices of T0 the strong continuity
of the semigroup at 0 need not imply D * = 0.
In general the sets ΰ0ηΰ*, D0nC*, C0nD*, and C0nC* may all be non

Positive 1-parameter semigroups 231
empty; see Lin [1982], Suzuki [1983]. It is simple to see that D* is the initially
dissipative part of the adjoint semigroup.
Lemma 1.2. D* a D0 holds when the Tt, (t > 0), are contractions, and also when
{Tut^ 0} is strongly continuous at t = 0 with || T0\\ = 1.
Proof. Take h = S0tlh' as above. In the first case ||A1C.||P = || TE(hlc*)\\p
= II Teh\\p -* ||A||p, (ε -* 0), implies h = hlc*. In the second case T£(h\c*) = T£h
tends in norm to T0h = A and to T0(hlc*). Hence ||А1с*||р = || Г0(А1с*)||р = ||A||P.
Thus h must vanish οηΰ*. α
The decomposition of Ω into C0 and D0 has been introduced by Akcoglu-Chacon
[1970], and that into C* and D* by Sato [1978b].
2. Positive additive processes. The following "differentiation theorem"
(Akcoglu-Krengel [1979]) unifies and extends several previous local ergodic
theorems:
Theorem 1.3. Let {Tt, t > 0} be a locally bounded strongly continuous semigroup
of positive linear operators in Lp, (1 rg ρ < oo), andF — {Fv t > 0} a positive addi-
tiveprocess in Lp with \\Ft\\p -> 0, (/ -> 0 + 0). Then Q-limt_+0 + 0 Γ1 Ft exists a. e. in
С* и D0for each countable dense subset QofR+. In particular, the Q-limits exist
a.e. in Ω under the additional assumption that the Tt are contractions.
The proof of this theorem will be obtained after several lemmas. We first define,
for / > 0, a class 0>t = ^t{F) of Lp-functions as follows: fe 0>t{F) if and only if
there exist two numbers r > 0 and s > 0 and an integer η ^ 0 and functions
fo>fu -"Jn^L+P such that
(i) r + ns rg /;
(ii) Tsm-Fr^ Σ ΤΓΊι form = 0,1,..·,л, and
Г i = 0
(iu) /= Σ/ι.
i = 0
In order to understand the intuitive meaning of this definition the reader may like
to recall the filling scheme of Chacon-Ornstein. If a hole/is filled with starting
m
mass g by repeated applications of 7^, we obtain Tsmg7> Σ Tsm~ (fi9 where/f is the
i = 0
part of the hole filled with a portion of Tjg. If the additive process is of the form Ft
= S0ttg we actually have a density g at the beginning. But in general we have to
replace it by r~l Fr. Thus, the functions fe £Pt{F) are the holes we can fill up to
time / if we reserve a short initial interval of length r in order to define an
"approximate density" r~l Fr = Fr with which we want to do the job.

232 Local ergodic theorems
Lemma 1.4. If fe 0>t{F) then S0tUfu Ft+ufor each u>0.
m
Proof. Let/=/0 + .··+/„ with Σ ί""'/ι й TsmFr for m = 0,..., η and r + sn
й t. Then i=0
η и η u + (n — i)s
S0.J= Σ ί TJtda й Σ ί U-ώ
ί = 0 0 ΐ = 0 0
η—1 s m и η
й Σ ί τα Σ r_7W« + ί £ Σ 37" V«
m = 0 0 i = 0 0 i = 0
^ "Σ ί Ta(TsmF,)da + } Tx(TsnFr)da
m = 0 0 0
n —1 (m + l)s _ u + ns _
= Σ ί TxFrda+ J Тдаа
m = 0 ms ns
u + ns _ u + ns 1
= J TaFrda = J -(Fa+r-F^da
о о г
J r + ns + κ J m + bs J « + ns + r
= 7 ί W*-- ί №й- ί £<**
г г ' 0 'u + ns
= ^w + ns + r = *w + i· а
For /? = 1, sup {\Efdp. fe 0\{F)} can be used successfully to measure how much
we can fill into a set Ε up to time /. But for general ρ with 1 ^ ρ < oo we have to
use
!Pi(F)'= sup {a ^ 0: Ve > 03/e ^(F) with ||alE -f\\p< ε].
Note that 0 й g й fe 0>t (F) implies ge0>t (F). Hence, if α0 = ΨΕ (F), we can fill a
hole of size α01£ up to time / arbitrarily well, but no hole β1Ε with β > oc0.
It is clear that 0 й ΨΕ\Ρ) й 4*e(F) if t' й t. Hence
WE(F):= lim WE(F)^0
i->0 + 0
exists.
Lemma 1.5. Let h e L*, h > 0 a.e. and h = S0Ah'. Assume that A > а > 0 ял
Ее si and μ (Ε) > 0. ГАеи #и;еи ε > 0 there is an Ε' α Ε so that μ(Ε\Ε') < ε and
so that the additive process Η = {Ht\ t > 0} given be Ht = S0th has the property
that ΨΕ>, (Η) ^ α - ε for all Ε" α Ε'.
Proof The definition of h implies that Ht = t~1Ht tends to h in norm as / -* 0.
_ °°
Hence || \EHt — \Eh \\p -* 0. Find εί > 0 with Σ ε,- < ε and for each /choose /f > 0
so that t{ > ti+1 -* 0 and so that l=1
μ(^η{^^α-ε})<εί.

Positive 1-parameter semigroups 233
This uses the finiteness. of μ(Ε), which follows from A ^ al£. Let
E'= f] En{Hti>oc-s}.
Then μ{Ε\Ε') < ε, and if E" a E' then Hti \E„ > (α - ε) \Ε„ for all tt. This means
that (a — ε) \E„ e 0>t. (H) for all /i? because we can take r = tbn = 0,/0 = /7,. to fill
the hole (a - ε) 1£„*. Now (a - ε) ^ Ψε*»(Η) and this tends to ΨΕ>(Η). □
Lemma 1.6. Le/ £ с C*? 0 < μ^Ε), /Леи *F£(F) < oo.
Proof. We may assume μ (Ε) < oo because smaller holes are filled more easily. By
the local boundedness of the semigroup we have K-= sup {||7J||:0</^l}<oo.
Let δ := sup {M0,tlE||p: 1/2 й t й 1}· Ε с С* and 0 < μ(£) imply 5 > 0. Choose
an integer N ^ 1 so that TV-1 A]| l£||p < δ/2, and let t1 = (N-l· 1)_1 We shall even
prove the finiteness of TE^{F).
If for some α > 0 there exists an/e 0>tl (F) with || al£ —f\\p < ε <; 1, then there
exists r > 0, я > 0, и ^ 0 with r + ns rg /x and / is a sum of functions fh
η _
(i = 0,..., л), with Σ ΤΓ'Τί й TsnFr. There also exists aneeij with ||e||p < ε
i = 0
and al£ ^/+ e. We need an upper estimate for a. Now
«1e ^/+ « = Σ (7ί " ТГ'/д + Σ 2Г 7i + e
i = 0 i = 0
й i.(fi-1?-lfd + T„Fr + e.
i = 0
Hence
n 1
a*^o,iViilf; = Σ {So,NtiJi ~ Ts So,NtJi) "I So,Nti TsnFr + S0Ntie
i = o r
η J Nil
^ Σ 5Ό,(»-ο*/ΐ+~ ί Ta+nsFrdc<, + S0jNtie.
t = o г о
Using 7^+nsiv = i;+a+ws - /£+„ we can continue
η 1 Nti+sn + r
i = 0 f Nii+ns
= a^o,nslf; + S0tKSe + ENtl+ns+r + S0tNtle
Dividing by 7W1? we obtain
^ο,^Ιε ^ «— · _50§Μ1Ε+ —S0tNtie+ -^
and

234 Local ergodic theorems
*\\A0tNtl 1E\\P й ^ \\A0t„lE\\p + 2\\A0tNtle\\p + 2\\FX\\p.
The definition of δ and the inequalities 1/2 ^ Ntx ^ 1 now imply
аг^а^||1Е||р + 2Л:б + 2||/?1||р^а^+2Л:б + 2||/?1||р.
Hence α ^ 2δ~ι (2Kb + 2\\FL \\p). We have obtained an upper bound for α which
does not depend on/e ^tl(F). This means that ΨΕι(Ρ) is finite, а
In order to prove the a.e.-convergence of t~x Ft, (/-> 0 + 0), when t ranges
through a countable dense subset Q с U + \ {0}, we may assume that Q contains
the positive rationals. For t0 > 0 let Q(t0) denote the set {s e Q: 0 < s < t0}.
Lemma 1.7. Let F= {Ft} be a positive additive process as in theorem 13 and let
ge Up. If sup {(Fq — S0tqg): q e Q(t0)} > 0 on Ее s/, then for any ε > 0 there
exists an/e^io(iO so that ||/-glE||p<e.
Proof. Let ε > 0 be given. By the strong continuity of the semigroup we can find
finitely many rational qte Q(t0), (1 rg i ^ n), and an а > 0 so that the sets
Et*= En {{Fq. — S0q.g) > a} have the property that
! «Ήί)'·
E\{JE, Ч '
t=l
Let δ > 0 be so small that ||glH ||p < s/4n for all Η e sJ with μ (#) < <5. Choose Μ
so large that m ^ Μ implies \\Rqt(m)g — S0tqig\\p< a<51/pfor/ = 1,..., л. As the
^ are rational numbers we can find r > 0 and integers m1?..., mn ^ Μ so that ^
= rmf for all i. Let Д = Ein{(Fq. — Rq.(m^g) й 0}. Then Д is contained in
{\RqiQni)g — S0tqig\><x}. The choice of Μ implies αρμ(Β^)<αρδ and hence
η
\\giBi\\P<8/4n. Let£" = (J (2?ДД), then Hgl^'Hp < ε. It now suffices to show
i = l
that gl£, e 0}oCF). On £Д Д. we have
тг-ι (\ \
Fqi-Rqi(mi)g = r Д 27i-/?-gJ>0.
Hence, letting AT = max (m1? ra2,..., raj we have A> < /0 and
sup Σ Trj(-Fr-g)>0 onE'.
0<k^Kj=0 \Г )
We may therefore apply the construction of the Chacon-Ornstein filling scheme:
Lemma 3.7.1 asserts that there exist functions d0, dx,..., dK_x e L+p such that

Positive 1-parameter semigroups 235
Σ Trk-jd:uTrk-Fr fork = 0,...,K-l
j = 0 Г
and d-= d0 + dx + ... + dK_i =gon Ε'. We have de 0\O{F) and, because of
0£glE,£rf,alsoglE,e^F). α
Lemma 1.8. Let {Ц} and {Gt} be two positive additive processes with \\Ft\\p -> 0
resp. ||Gt\\p -* 0, (/ -* 0 + 0), аля?assume sup {(7^ — C?,): / e Q(t0)} > 0 от Ее si
with μ (Ε) < oo. Then, for each ε > 0, /Aere ejrato a set Ε' α Ε and α δ > О .шсА /Αα/
μ(Ε\Ε') < ε, and such that, for all r ^ <5,
sup {№- 50>tg): ί g β (ί0)} >0onF for all ge&r(G).
Proof We find finitely many t{ e (20o)? 1 ^ * ^ и, and an α > 0 such that if Et
η
= En {Fti - Gti > a} then μ(Ε\ (J Et) < ε/2.
i = l
Since G is strongly continuous and Gti+S decreases to Gt. when s decreases to 0,
there exists a δ > 0 such that the sets Д. = {Gff+i — C?it > a} have measure
μ(Βί)<ε/2η, (ι = 1, ...,/i). If 0 < r ^ <5 and ge ^(C?),'then ge^(C?). By
Lemma 1.4
^ - So.** ^i " <?ί|+, ^ W, " Gt) ~ (Gti+d - Gt) > 0 on Et\B{.
η
Therefore 2?' = (J (2?ДД) has the desired property, α
i = l
Lemma 1.9. Assume that sup {(Ft — Gt): t e Q(t0)} > 0 for all to>0 on a set
Ее si with μ(Ε)<οο. Then, for each ε > 0, there exists a set E' с Е with
μ{Ε\Ε') <sso that WE^(F) ^ WE»(G) whenever E" e si and E" с Е'.
00
Proof. We choose st > 0 with Σ ε* < β and also ^ > 0 with tt J, 0. By the previous
i = l
lemma we find for each i a set E[ and a number <5f > 0 such that μ{Ε\Ε[) < εί and
00
sup{(^- S0ttg): te Q(tt)} > 0 on E[ for all ge 0>6t(G). Let £' = f| E[. Then
i = l
μ(Ε\Ε') <ε. Let £"' cFbeafixed set. Let α = ΨΕ>(0). Given any ξ > 0 and ι
there exists a ge0*di(G) with ||g —alE"||p< £. Hence, as we have
sup {(i?-SO tg): teQ(td} >0 on £"', there exists <m fe 0>t.(F) with
II/- Wllp < ί· Now II/- alE„||p < 2f. As ξ was arbitrary >|ЦЛ ^ «, and
hence ΨΕ.. (F) = lim^β У *, (/0 ^ α = **V (<?)· □
Proof of theorem 1.3. For 0 < 5 < t, Ft = Fs + 7^_s. By ||/J ||p -* 0, (s -* 0) and
Lemma 1.1, we see that i^ = 0 on D0. Thus, the convergence of t~x Ft on Z>0 is
trivial.

236 Local ergodic theorems
Let h! e Up,h! > Oa.e., h = S01h\ and#, = SO , A as in Lemma 1.5. It follows
from
StAh'uTshuS0tl+th\ (OusuO
that Sttlh' й t~lHt S S0tl+th\ and therefore that Q-limt'1^ = h exists a.e..
Assume theorem 1.3 fails. Then 2-Нт^0 + 0^Д-1 fails to exist on some
setEejtf with 0 < μ (Ε) < oo and Ε a C* η {h > 0}. Passing to a smaller set with
these properties we may assume that there exist numbers α > 0 and 0 < βγ < β2
so that h > en on Ε and
0 й £-Ит inf FtЦ'1 <βγ<β2< β-Ит sup FtH~l
i->0 + 0 i->0 + 0
on E. Using Lemma 1.5 with ε = α/2 we find E' с Ε with 0 < μ(Ε') so that
ΨΕ»(Η) > α/2 > 0 for all E" с Е'. Observe that
sup (/?-02£(мЛ)>О and SUP (^i5O,iA-i?)>0
ieQ(io) ' ieQ(io)
on £" for all /0 > 0. Using Lemma 1.9 we can find E[ cz E' and £2 с £" so that
**V(^) ^ Ve-WiB) = β2ΨΕ»{Η) and /i1 УЕ„(Л) ^ yE„(F) for all
E" cz E*--= Е[слЕ'2. Choosing the epsilons in the application of Lemma 1.9
small we may assume μ(Ε*) > 0. For E" = E* we get β1 ΨΕ*(Η) ^ β2 «F£*(#)
and this contradicts βχ<β2 because Lemma 1.5 and Lemma 1.6 imply
0 < ΨΕ*(Η) < oo. This completes the proof of the theorem, α
3. Non positive additive processes. If {Tt, t > 0} is a semigroup of operators in Lp
as in theorem 1.3, the a.e.-convergence of A0tf follows also for non positive
fe Lp because of A0ftf= A0ftf+ - A0ftf~.
We now discuss a similar but less obvious decomposition for a wider class of
additive processes; see Akcoglu-Krengel [1978], [1979].
An additive process satisfying
(1.1) b(F)i=supr1\\Ft\\p<cx).
0<i
will be called linearly bounded.
Theorem 1.10. Let {Tt, t > 0} be a locally bounded strongly continuous semigroup
of positive linear operators in Lp,{\ ^ ρ < oo). The family of all linearly bounded
additive processes is a vector lattice. In particular, any linearly bounded additive
process F is of the form Ft = Ft{1) — Ft{2\ where F{1\ F{2) are positive linearly
bounded additive processes.
Proof For dyadic rational t = k2~n put
Ft(n,i) = У Ti-n(F2-ny and Ft^ = V Ti-n(F2-ny.
j=0 j=0

Positive 1-parameter semigroups 237
From
(л2-") = {F2-(n+i) + T2- (n+ i)F2- (n+ i))
it follows that Ft(nA) й Ft(n+U1). Similarly, F^2) й Ft(n+1>2). For dyadic rational
/ > 0 put Ft(i) = limm_aoFt{mti\ The additivity and the linear boundedness can be
verified in a straightforward way for dyadic rationals.
The new processes satisfy \\Ft{i) — Fs{i)\\p rg b{F) -\t — s\. Therefore we can
define Ft(i) for general t > 0 by F^ = limsttFi(i), where s ranges through the dyadic
rationals. The additivity and linear boundedness of the processes Ft(l) follow by
continuity. Clearly, F{1) = FvO. α
If {37, / ^ 0} is strongly continuous at f = 0 with T0 = /, then the
decomposition of Ft = Sotf has the form Ft{1) = S0tf+. Theorem 1.10 is also related
to the decomposition of functions of bounded variation: Let Ω be the real line
with μ = Lebesgue measure. For a measurable/: Ω = R -* R define Ft by i?(co)
= /(ω + i) —/(ω). The translation operator Ttg(a>) = g(a> + /) is a contraction
in Lx, For contractions the supremum in (1.1) is a limit as / -* 0 + 0. We define
the essential total variation ||/||β88.ι.ν. °f/by
ll/lles,,v.= Hm U\f(co + ή-/(ω)\μ(<Ιω) = b(F).
Recall that the total variation \\f ||t v of a function/': R -+ R is the supremum of
η
all sums Σ I/'(A) —/'(A-i) I where the sup is taken over all t0 < tx < ... < tn in
ί = ί
R. If/' has finite total variation,/' is the difference и — ν of two bounded
increasing functions u, ν and ||/||t.v. = || w llt.v. + IMIt.v.· F°r increasing functions the
essential total variation coincides with the total variation because
\ +oo \ b
- J |u(x + /) — u{x)|dx = lim - J (u(x + 0 — u{x))dx
t — oo a-> — oo * a
b-> + oo
(\ b + t \ a + t Ί
lim <- J u(x)dx j" u(x)dx>
a-+ — oo [t b t a J
= sup u(x) — inf u(x) = ||w||t.v..
It follows that ||/||ess.t.v. й \\f llt.v. holds for all measurable/and for all/' with
/' =/a.e.. The terminology "essential total variation", in analogy to the
"essential supremum" as the minimal supremum in the equivalence class, is justified by
the following result of Akcoglu and Krengel [1978]: For any measurable/with
ll/lless.t.v. < °o there exists an/' with/' =/a.e. and ||/||eM.t.v. = ||/'llt.v.· In this
example the decomposition of Ft has the form i^(1)(co) = u(a> + /) — и(оэ),
Ft(2\co) = ν(ω + ή-ν(ω).

238 Local ergodic theorems
Stadje [1984] has given a different proof and observed that one can take/'(x)
x + t
= lim t~l J f(u)du.
ί->0 + 0 χ
4. Continuity at the origin. By a theorem of Dunford, a semigroup {Tt, t > 0} of
bounded linear operators in a Banach space X is strongly continuous, when the
map / -* Ttf is strongly measurable for all/e X; see § 1.6. Thus, the strong
continuity of the semigroup at t> 0 is a rather weak assumption. For the questions
of convergence with / -* 0 + 0 considered here, it is of interest to know when
there exists a bounded linear operator T0 for which {Tu t ^ 0} is a semigroup
which is strongly continuous also at t = 0. We shall then say that {Tt, t > 0} is
strongly continuous at / = 0.
The following theorem (R. Sato [1975a]) implies that for X = Lp, 1 <p < oo, a
locally bounded semigroup strongly continuous at / > 0 is also strongly
continuous at / = 0:
Theorem 1.11. Let {Tt, t>0} be a semigroup of bounded linear operators on a
Banach space X which is strongly continuous at t > 0. Assume that, for eachfe X,
{Ttf: 0 < / < 1} is weakly sequentially compact. Then {Tt,t>0} is also strongly
continuous at t — 0. The same conclusion holds if, for eachfe X, {A0tf 0 < t < 1}
is weakly sequentially compact.
Proof. As each set {Ttf: 0 < / < 1} is norm bounded, the uniform boundedness
principle implies that the semigroup is locally bounded. It follows that
(1.2) {fe X: lim || TJ-f\\ = 0} = cl{ [j TtX).
ί->0 + 0 ί>0
For any/there exists a closed separable subspace Xf of X containing/such that
TtXf с 3C/? (/ > 0). Therefore we may assume without loss of generality that X is
separable. Let {fh i ^ 1} be a dense subset of X. By the diagonal argument there
exists a strictly decreasing sequence /„ -* 0 such that Ttnf converges weakly for all
/. As the semigroup is locally bounded it follows that ^/converges weakly for all
fe X. Let T0f denote the limit. Clearly T0 is a linear operator and it is bounded
because the semigroup is locally bounded. It is straightforward to verify TtT0
= T0 Tt = Tt for / ^ 0. Hence, putting fx = T0f we see that each fe X can be
written as a sum/ = /1 +f2 with T0fx =ft and T0f2 = 0.
We claim that Ttfx converges strongly to/x for / -* 0 + 0. Otherwise, by (1.2)
and Hahn-Banach, there exists an h e X* with 0 φ </l5 A> and with (Ttf h} = 0
for all t > 0 and/e X, a contradiction to/i = w-lim Ttnf Thus, T0 is the limit in
the strong operator topology of the Tt. The proof of the second assertion is the
same, α
Corollary 1.12 (Akcoglu-Chacon [1970]). If{Tut> 0} is a locally bounded semi-

Positive 1-parameter semigroups 239
group of positive linear operators in Lu strongly continuous at / > 0, and ifD0 is
empty, then {Tt,t>Q} is strongly continuous at t = 0.
Proof For strictly positive h! e Luh — S0A h is strictly positive. Anyfe ϋγ is the
limit of/(n) =/л nh. Now 0 й Ttf(n) й nSttt+lhf й n(h + 7\A), (0 < t < 1), and
the local boundedness of semigroup imply that {Ttf 0 < / < 1} is uniformly inte-
grable and hence weakly sequentially compact, α
The following example of Akcoglu-Chacon shows that for D0 Φ 0 continuity at 0
need not hold:
Take i3 = Ru {ω0} with ω0 φ R and let μ be the Lebesgue-measure on U and
μ{ω0} = 1. For/e Lx and / > 0 put (Ttf) (ω0) = 0 and
(Ttf) (ω) =/(ω0) (2π/Γ1/2 exp {- ω2/2ή
+ f (2π/Γ1/2 exp(- (ω - η)2/2ί)ηη)μ(άη),
us
for ω Φ ω0. If/= 1{ωο}? then ^40>ε/converges to 0 a.e. on Ω for ε -> 0.
Consequently, the strong continuity of {37, / ^ 0} at / = 0 would imply T0f= 0. But
|| Ttf\\x = 1 for all / > 0. Heuristically, the contractions Tt are given by Brownian
motion plus a contribution of the mass/(co0), which is transformed as if it was
located in {0} at time 0.
If a semigroup {Tt, 12> 0} is strongly continuous at / = 0, r0/is the limit of
A0ef in the local ergodic theorem. If the operators Tt are positive contractions in
Ll9 and D0 is empty, Г0/тау be described as a conditional expectation by
passing to an appropriate reference measure: Let ν be the measure having density
h with respect to μ. By the isomorphism Lx (μ) э/ <-> f/h e Lx (v) the semigroup
induces a semigroup {Tt\ 12> 0} in Lx(v): 2^'g = A-1 !ζ(Α -g). This new
semigroup has the property 7J1 = 1. It is known that a positive contraction Tq
in Li(v) with (Γό)2 = Tq and ГОИ =11 is a conditional expectation operator.
(See Ando [1966], or Neveu [1965: 123]). Call A e si initially invariant if
Ίο 1a — 1a· The initially invariant sets form a σ-algebra </0, and we have T^g
= Ev(g\ J0). Thus, we may say that using a suitable reference measure ν the limit
in the local ergodic theorem is a conditional expectation just as in Birkhoff 's
theorem. This "identification of the limit" is due to Lin [1972]. It is not hard to
show that A is initially invariant if and only if there is some f e L\ (μ) with A
= {/> 0} and J^ 7^/V//* -ι- H/IU.
The case, where T0 is the identity operator in Lx may merit special attention. It
is characterized in the following theorem of Lin [1972]:
Theorem 1.13. Let {Tt,t>Q} be a semigroup of positive contractions in Ll9
strongly continuous at t > 0. Then the following statements are equivalent:
(i) For every fe Lx we have limt^0 + 0A0tf = fa.e.;

240 Local ergodic theorems
(ii) The Tt converge to the identity in the strong operator topology (t -> 0 + 0);
(Hi) The Tt converge to the identity in the weak operator topology (t -> 0 + 0);
(iv) μ{Α) > 0 implies Нти0+0 II V \A\L = 1.
Proof. Clearly (i) implies D0 = 0, so that (ii) follows from corollary 1.12.
Conversely (ii) implies (i) by theorem 1.3.
(i) => (iv): Take/e L+x with \\f\\x = 1 = \Α/άμ. Using Fatou's lemma we find
1 = </, 1 a> = <Hm A0tgf, \АУ й Ит inf <Λ0,ε/? 1л>
ε->0 ε->0
^ lim inf β"1 J II3?* 1^ Поо Л ^ 1 ·
ε-+0 Ο
Now (iv) follows because || Tt* 1A Ц^ is a decreasing function of/. The same/may
be used to prove (iii) => (iv):
1 = Шл> = Ит <^/,l^> = Ит <f,T* 1Л>
ε->0 ε->0
^H/Hilimll^l^lL^l.
ε->0
As (ii) => (iii) is trivial, it remains to show (iv) => (ii):
Let L be the closed subspace of X = Lx described in (1.2). If (ii) fails then
L Φ Lv Then by Hahn-Banach there exists dig e L^ = £* withg Φ 0 and </, g>
= 0 for/e L. In particular, <2^, g> = 0, (t > 0,/e Lx). Hence ?;*g = 0, (t > 0).
We may assume HgH^ rg 1. For<5 > Oput^ = {ω:#(ω) ^ <5}. —H ^ ^ ^ H yields
1 +g^(l + d)lA<md
(l + S)Tt*lAu Tt*(i +g) = j;*H1.
This implies || ?;* 1 J| да й 1/(1 + <5) for all t > 0 and, by (iv), /ι(Λ) = 0. As δ > 0
was arbitrary we obtain μ^>0) = 0. Applying the same argument to — g we
also obtain μ^ <0) = 0. But g = 0 contradicts the choice of g. D
5. The local ergodic theorem in L^. Let {Tt, t > 0} be a strongly continuous locally
bounded semigroup of positive linear operators in Lv If A$ftg converges a.e. (as
/ -* 0) for all ge L^, then </, ^4*,г£) converges for all/e Lx by Lebesgue's
bounded convergence theorem. Hence {A0itf: 0 < / rg 1} is weakly sequentially
compact in Lt for all/e Ll9 and {Tt91 > 0} is strongly continuous at / = 0 by
theorem 1.11. We now prove that conversely the strong continuity at / = 0
implies the a.e.-convergence of A% tg, (t -> 0), for all g e L^. This was shown in the
case Ω = C0 by Sato [1978b], and in general by Lin [1982]. (For the case of null
preserving transformations; see Krengel [1969a]).
Theorem 1.14. If{Tt, t ^ 0} is a strongly continuous locally bounded semigroup of
positive linear operators in Lx andg e L^, then A$tg converges a.e. as t -> 0 + 0.

Positive 1-parameter semigroups 241
Proof. The proof is a reduction to the local ergodic theorem in L{. For any/e Lx
we have S0tf= 0 on D0. Therefore A%tg is = 0 for g e L^Dq), and it is enough
to prove the convergence of A%tg for g e ^(С0).
We first prove the convergence on C0. Lx (C0) is invariant under {Tt, t ^ 0}. Let
Rt be the restriction of Tt to Lx (C0). Then, for/e Lx (C0), g e L^Cq), we have
</, JVi> = W,£> = Ж#> = </, 3?*g>, so that Jtfg = T*g on C0. We
may therefore assume Ω = C0 for this part of the proof.
As {Tu t ^ 0} is strongly continuous there exist Μ < oo and α > 0 with || 2^||
й Meta/2, (see e.g. Yosida [1974: 232]). If h e Lx is strictly positive in Ω = C0
then h — S0tlh' and hence also
u= ]e-"Tth'dt
о
is strictly positive in Ω. The choice of α implies ue Lx. For / ^ 0,
00
Ttu= J e-atTt+shfdsueatu.
о
Hence, for / ^ 0 and ge L^, we see that
§(e~CίtTt*g)udμ — ^β'^^Τ^άμ ^ Jgwrf/i,
so that e-ai 7^* is a contraction in Lx(Ω, .я/, w · μ). Forfe L^, ge L^ and f -* 0,
\Αβ-«Τ*ξ)ηάμ = e"" J W« M)g<//i -
By approximation this shows that e~atTt* is weakly continuous at 0 in
Lx (Ω, s&\ u\x) and hence also strongly continuous. As L^ is contained in
ε
Lt(Q9 si, w μ) theorem 1.3 implies the convergence a.e. of ε-1 §e~atTt*gdt,
о
(ε -* 0). (Actually the simpler special case of continuous semigroups of positive
Li -contractions is enough). As e~at tends to 1 for / -> 0, we can conclude that
ε ε
also ε-1 J Tt*gdt converges a.e.. As the integrals S0ttf and J Tt*gdt can be ob-
o ' о
tained as limits of approximating Riemann sums in Lx (μ) and in Lx (u · μ) it is
ε
simple to check J Tt*gdt = S$tBg and the proof of the convergence on C0 is
complete. °
Now let Q be the set of positive rationals, ge L^(C), and let g
= e-Hmsupe_,0^Jfl!g, g = Q-\imix\i&^0A%^g. H^IL й Меш12 implies WgW^
й \\g\L й M\\g\\„ < сю. Using T0*(gl v?2v...v?„)^ (Tfgl) ν ... ν (T0*gn)
and a representation of g as limit of suprema of averages A $tft g it may be seen that
T0*g ^ β-lim sup - } Г0* T*gdt = g.
ε-0 β Ο

242 Local ergodic theorems
Similarly Γ0* g <; g. Hence Г0* (g — g) ^ g — g (^ 0). But g — g has support in D0.
Therefore Г0* 1D = 0 yields T0* (g-g) = 0, and g = g. α
Notes
The local ergodic theorem for a strongly continuous semigroup {Tt, t ^ 0} of positive
contractions in Lu obtained independently by Krengel [1969b] and Ornstein [1970], has
a much simpler proof than theorem 1.3; see the proof of theorem 2.4. Akcoglu and Chacon
[1970] showed that the continuity at t = 0 was not needed. Fong and Sucheston [1971]
and Kubokawa [1972] replaced the contraction hypothesis by weaker assumptions.
Kubokawa [1974] proved the local ergodic theorem for a locally bounded strongly
continuous semigroup {Tu t ^ 0} of positive operators in Lp, (1 <Ξ/? < oo), with T0 = L
Theorem 1.3 unifies these results and extends them to additive processes. The proof uses
ideas of Akcoglu-Chacon and of Kubokawa. Sato [1978 b] proved local ergodic theorems
in which the assumption of local boundedness is replaced by the assumption that for each
fe Lp the vector valued function t -* Ttfis Lebesgue integrable over every finite interval.
Emilion [1984b] has a (nontrivial) argument showing that theorem 1.3 (for/? = 1) can
be deduced from the special case due to Krengel and Ornstein. Moreover, he shows for Ft
^ 0, that/= £Mim t'lFt is the largest function/with Ff ^ 50ff/, (t > 0).
Sato [1978b] has given an example of a strongly continuous semigroup {Tt; t ^ 0} of
positive operators in Ll with Tt1 = 1) and || Т(\\х = 1 + δ for which A0sf fails to converge
а.е.. Thus, convergence need not hold on D*; see also Akcoglu and Krengel [1979a].
If X is a reflexive Banach space and {Ft, t > 0} a linearly bounded additive process for a
strongly continuous locally bounded semigroup {Tt,t>0}, then Ft is of the form Ft
= S0ttf; Akcoglu and Krengel [1979].
For a technique of deducing local ergodic theorems in Lp, (1 < ρ <Ξ oo) from theorems in
L, see McGrath [1976], [1976a], Lin [1982], Sato [1978b], Emilion [1984d].
Feyel [1982] proved the following superadditive differentiation theorem:
Theorem 1.15. Let {Tt,t>0}bea strongly continuous semigroup of positive contractions in
Lv LetF= {Ft,t>0} satisfy
(V ΙΙ^ΓΊΙι^ Mt for some Μ< oo, and
(ii) Ft+s^Ft + TtFs,(t,s>0).
Then g-limf_>0 + 0 t~1Ft exists а.е..
Akcoglu-Krengel [1978] had proved this for Markovian {7;} and WF^^Mt,
(t ^ 1), (holding for F^ 0); see also Emilion-Hachem [1982], and Akcoglu [1983].
Again, generalizations in Lp follow from theorem 1.15; see Kataoka, Sato, Suzuki
[1983], Emilion [1982].
A way of expressing the existence a.e. of g-limits for arbitrary countable dense Q с R+ is
to use essential limits; see Feyel [1982]:
For a family (f) of equivalence classes of measurable functions and la U+ let
ess-sup {f, t g /} be the supremum in the set of equivalence classes. (It is the supremum
over a suitable countable subset /0 <= /; see Neveu [1965], § 2.4.) Put ess-limsup^o + o/,
= ess-infs>0 (ess-sup {f: 0 < t < s}). ess-lim is said to exist if ess-limsup = ess-liminf.

Non positive and multiparameter semigroups 243
§ 7.2 Local ergodic theorems for multiparameter and non positive
semigroups, and for vector valued functions
1. Multiparameter local ergodic theorems. Pd denotes the set of vectors и
= {uuu2,...,ud)eUdwithщ > 0, (ι = 1,..., d), and Ρ°d denotes {0} иPd9 where
0 = (0, 0,..., 0). и < ν, (и rg ν), means that щ < vh {щ rg v^, for all i. Let [w, v\
= {/eRd:i/^/^D}, e = (1,1,..., 1), and π{μ) = ux- u2 ...ud. A semigroup
ρ = {Tu:ue Pd} of linear operators in Lp is called locally bounded if
sup {|| Tu\\: и < ν) is finite for one and hence for all ν > 0. If У is strongly
continuous and locally bounded we may define the integrals
Su.vf=]Ttfdt= f Ttfdt
и [и, ν]
as in the 1-parameter case (§1.6) by approximating Riemann sums or by a
suitable choice of the representatives. Put Auv = π(ν — u)~xSUtV9 and Шр
= {A0vg: 0 < v9 g e Lp}. As usual и -* 0 + 0 means that щ -* 0 + 0 for all i.
Lemma 2.1. Let 2Г be a locally bounded strongly continuous semigroup of linear
operators in Lp — Lp{ii, si\ μ), (1 Sp < °o)· Then A0uf converges a.e. to f as
u^0 + 0forallfe9Rp.
Proof. Let λ denote the Lebesgue measure in Pd. For a suitable choice of the
representatives \Tsg(co)\p is μ χ Я-integrable on Ω χ [0, и]. Hence |Z^(co)|is
μ χ Я-integrable on Α χ [0, и] for μ (A) < οο. For 0 < w < ν consider
^,υ(ω)= ί \Tag((o)\ds.
[0,v + w]\[w,v]
When w decreases to 0 the sets [0, ν + w] \ [w, υ] decrease to a Λ-nullset and
dwv(o>) decreases a.e. to 0. For Org/rgwrgwwe have
\TtS0tVg-S0fVg\ udWfV.
Hence \A0uS0tVg — S0tOg\ й dwv. If и tends to 0 we obtain
lim sup \A0tUS0fVg - S0tVg\ й dWfV.
м->0 + 0
Because of dwv -* 0, (w -* 0) the assertion of the lemma is proved for S0tVg, and
hence for/ = A0tOg. α
To derive a local ergodic theorem we also need a maximal inequality. First notice
that Lemma 1.6.1 together with Theorem 6.3.4 yields the following discrete
parameter inequality:
Lemma 2.2. There exists α constant c(d) > 0 with the following property:
If Tl9 T2,..., Td are commuting Lx — L^-contractions then

244 Local ergodic theorems
/φυρμ,,/Ί >«);£— J \ί\άμ
n^l α {|2/|>α}
holds for all oc > 0 andfe Lv
Applying this to the operators 72-*<o,o,...,i,o,..) and to the function Г0/опе
obtains an inequality of Dunford-Schwartz by discrete approximation:
Lemma 2.3. If {Tu:ueP%} is a semigroup of L1 — L^-contractions which is
strongly continuous (also at и = 0!), then
μ( sup \A^J\>ol)u— f \Τ0/\άμ
0<ε<οο α {|2Γο/|>«}
holds for all a > 0 andfe Lv
The detailed argument is the same as in the proof of Theorem 1.6.11. It is now
easy to derive the first local ergodic theorem of Terrell [1970]; (see also Terrell
[1972]):
Theorem 2.4. Let {Tu:ue P°} be a strongly continuous semigroup of Lx — L^-
contractions, then lime^0A0eef= T0fa.e. for all fe Lx.
Proof We may assume T0f = f because A0tBef = A0tfteT0f. Now fv = A0tOf
tends in norm to T0f since the semigroup is continuous at 0. By lemma 2.1
lim sup \A0ttef- T0f\ й Ит sup \A0tBef- A^eefv +fv - T0f\
ε->0 + 0 ε->0 + 0
U\f0-T0f\+ sup \A0,„(T0f-fv)\.
0<ε<οο
Both terms can be made small by a small choice of ν using \\fv — T0f\\x -* 0 and
lemma 2.3. α
Let us briefly discuss more general averages. For measurable В with
0 <λ(Β) <οο put
A{BJ) = X{B)-'\TJdu.
в
Essentially the same proof yields convergence a.e. of A (Be9f) for/e L1? if {Βε} is
a family of sets with Βε c [0, &e~\ and there is some β > 0 with λ(Βε) ^ βεά.
In the special case of translations Tuf{x) = f(x + u) in Ud the question of
convergence /Ua.e. of averages A (B(q),f) for suitable families {B{q)} has been
investigated very thoroughly starting with a famous theorem of Lebesgue [1910]: He
showed that A{CE,f) converges Я-а.е. for Λ-integrable/if Cg is
d
[ — ее, + ге~] or {χ e Ud: Σ xf ^ ε}. (It is easy to see that this classical theorem
i=l

Non positive and multiparameter semigroups 245
follows from the assertion above about convergence of A(Be,f)). Saks [1934],
and Busemann and Feller [1934] have shown that A0uf may diverge for Λ-
integrable / for и -* 0 + 0.
The next theorem gives a sufficient condition for unrestrictedconvergence, due
to Zygmund (in the translation case), and to McGrath [1976]; (see also theorem
2.13):
Theorem 2.5. For some ρ with 1 <p < oo let {Tu\ ue Ρ °} be a strongly continuous
semigroup of positive contractions in Lp. ThenA0 uf converges a.e. to T0fasu -> 0
+ 0/or/eLp.
Proof, Lemma 2.1 yields convergence for/e 90ϊρ, and by the strong continuity 90ϊρ
is dense in T0LP. Now the d-dimensional extension of the maximal inequality of
Akcoglu (Theorem 6.1.2) is used to prove an analogous inequality for the
continuous parameter case by discrete approximation. Then the proof can be
completed as above, α
Theorem 2.6 (Terrell [1970]). If { Tu: и e Ρ J} is a strongly continuous semigroup of
positive contractions in Lx andfe Llf then A0eef converges a.e.for ε -> 0 + 0.
Proof (See McGrath [1976]). For some α > 0 define a new semigroup by T/f
= e-«^+t2 + ~'+td)Ttf and define A'0tU and S'0tU with T/ as A0tU and S0tU were
defined with Tt. Because of <?~a(il+i2+-+id) -* 1, (/ -* 0), it is enough to prove the
a.e.-convergence of A'QtZef.
For some strictly positive h'e Lt put C0 = {T0h' >0}. For /;> 0, T0f
= lim,,..^ T0(nh' л/). Thus r0/vanishes for all/^ 0 in Cc0 and this implies that
Ttf= T0 Ttf vanishes in Cc0 for all / and all/e Lv Because of A0eef= A0ee T0f
we may assume Ω = C0 for the rest of the proof. Now the function
A= J Tt'h'dt
is strictly positive in Ω and integrable. Moreover
(2.1) T;h= J Tt'h'dtuh
for all u. The measure ν with density h with respect to μ is equivalent with μ. Put
Tt"g = (Tt'(g · h))/h forge Lx (v). Using (2.1) it is easy to see that {Tt"\ и е Pj} is
a strongly continuous semigroup of positive Lx — L^-contractions in Lx (v). The
proof is completed by an application of theorem 2.4 to A'a^g = A'0 eef with
g=f/h. α
2. Non positive operators. For the proof of a local ergodic theorem for a non
positive semigroup {Tt} of contractions in Lx we begin with the construction of a
minimal semigroup {7^} of positive contractions dominating {71}.

246 Local ergodic theorems
The linear modulus | T\ of a bounded linear operator Tin a real or complex space
Lu as discussed in § 4.1, is the minimal linear operator with | Tg\ rg | T\ \g\ for all
ge Lx. The operators | Tt\ in general do not form a semigroup: the properties of
the linear modulus only imply | Tt+S\ ^ | Tt\ \ Ts\. Therefore ft has to be defined
differently. Put T, = |?;|.
Theorem 2.7 (Kubokawa-Kipnis). Let {Tt: / > 0} be a semigroup of contractions
in Lx. There exists a semigroup {Tt\ t > 0} of positive contractions with the
following properties:
(a) \Ttg\ ^ ft\g\ holds for all ge Lx and for all t > 0,
(b) if{Ut: t > 0} is a semigroup of contractions such that \Ttg\ ^ Ut\g\ holds for
all t>0 and for all g e Lu then Tt ^ Ut.
If{Tt:t>0} is strongly continuous, then {Tt: t > 0} is strongly continuous. The
same result holds for semigroups defined also at t — 0.
Proof For t > 0 let % denote the family of all finite subdivisions 0
= s0 < sx < s2 < ... < sn = t of [0, t\. If s = (X) and s' = (s'j) are two elements of
% we write s < s' if s' is a refinement of s. With this partial order % is an
increasingly filtered set. For/e Ux put
Φ(8,Λ = τ51·τ52_5ι...τ5η_5η_1/.
It follows from Ία+β <; ΎαΎβ that s < s' implies Φ(δ,/) 5Ξ Φ{$',ί)· As all operators
T„ are contractions, ||<P(s,./^lli = ll/lli and % can be defined by
Ttf= sup {Φ(8,/): se 9t} = lim i>(s,/),
se§t
using the monotone convergence theorem for increasingly filtered families. It
is simple to check Tt(fx +/2) = 7^ + Ttfl9 Ttaf= aftf (a ^ 0), and || Ttf\\x
S H/lli· Therefore 7^can be defined for all/e Lx as a positive contraction in Lx.
If 0 = s0 < ... < sn = / and 0 = s'0 < · · · < s'm = *'are subdivisions of [0, /] and
[0, /'], then 0 = s0 < ... < sn = sn + s'0 < sn + s\ < ... < sn + ^ = / + t' is a
subdivision of [0, / + /']. Conversely every subdivision of [0, t + i'] which is fine
enough to contain / is of this form. This yields Tt+t, = ft% and {ft: t > 0} is a
semigroup.
The proof of the assertions (a) and (b) is straightforward from the construction
of?;.
If the semigroup is also defined for / = 0, then T0Tt = Tt implies T0Tr ^ Tt. On
the other hand \\T0Ttf\\x й II VHi f°r/^ 0. Hence Т0Т, = Ti? (t ^ 0). We may
then put f0 = T0 and use 4>'(s,/) = TSlTS2_Sl... TSn_Sn_ tT0/instead of *(s,/) to
define 7J for ί > 0. Then one can proceed as above.
It remains to prove the assertion on strong continuity. As the Tu are
contractions it is enough to prove || Tsf— Ttf\\x -* Ofors -* t + 0and/^ 0. By the proof
of theorem 4.1.1 there exist for any ε > 0 finitely many gx, g2,..., gk with | & | ^ /
and ||TJ- maXi <j<fe | TJ£iI Hi < ε. Hence

Non positive and multiparameter semigroups 247
ΙΚΐΖ-ΊίΛΊΙι^Ιΐσ,/- max \Ttgi\r\\i + s
g ||(max \Tsgi\- max |^|)ΊΙι + β < 2ε
when s is so close to / that || 7^· — Т^^\х is smaller than ε/к. As ε was arbitrary
lim ||(Ts/-TjTHi = 0, {fe L\).
s->i + 0
Using Ts_tTt/^Ts/and Ts_tTt/=(Ts_iTt/-TI^+-(Ts_,Tt/-Ti/)-^Tt/
we obtain
\\<Xf-Xf)4i й \\<X-t\f-\f)4i
й \\<X-t%f-τ,ΛΊΙι + ΙΙΐ-,ΐί/ΙΙι - WXfh
u\\<xf-xn-\\i-+o.
Hence
(2.2) lim i|Tsy-Tt/||1 = 0.
s-+t + 0
For each s = (st) in ^ with / > 0 and fe L\
\MJ Tsj) S (TtJ — T(s_i)+SlTS2_Sl . ..TSn_Sn_lj) .
For given ε > 0 there exists se^ with
ιιί;/-τ51τ52-51...τ5η-5η.1/ιι1<ε.
Then
IK?/- ίΛΊΙι^ II?/-T(s_i)+SlTS2_Sl...TSn_Sn_1/||1
=£ + ll(TSl — T(s_i)+Sl)TS2_Sl. ..TSn_Sfi_1/||1
and lim \\{TJ- Tsf) + \\1 = 0 follows from (2.2). Because of J Τ,/άμ ^ J ?/<//i
s->t + 0
this even implies || Ttf— Tsf\\1 -* 0 for 5 -* / + 0. The proof for a semigroup
{Ти t ^ 0} defined also at / = 0 is analogous, α
A combination of this result with the local ergodic theorem for positive
contractions now yields a local ergodic theorem for non positive contractions:
Theorem 2.8. Let {Tt, t ^ 0} be a strongly continuous semigroup of contractions in
Llf then A0ef converges а.е., (ε -> 0 + 0),for all fe Lv
Proof. We may assume fe T0LV For any ε > 0 there exists a small и > 0 with
II/— A0tUf\\1 < ε. Convergence of A0tft(A0tUf) for ε -* 0 follows from Lemma
2.1. Put g =f-A0,uf and A0tSg = ε"1 } ftgdt. Then
0

248 Local ergodic theorems
lim sup \A0ttg\ й Hm sup |io,e|g| I.
ε->0 ε->0
But by the local theorem for positive contractions (Theorem 1.3 or Theorem 2.6)
the second lim sup is a limit and the integral of the limit is ^ \\g\\i < e. As ε was
arbitrary the theorem is proved, α
3. Local ergodic theorems for vector valued functions. We now study local ergodic
theorems for semigroups of operators acting in the space Lx (Ω, si, μ, X) of Boch-
ner integrable functions on Ω taking values in a Banach space X with norm | · |. As
many arguments are similar to the real valued case the proofs will be only
sketched or even omitted. We shall use some notations and results from §4.3.
If {Tu:ue [0, oo)d} is a strongly continuous semigroup of contractions in
Lx (Ω, si, μ, X) = L\ the strong continuity may be used as in § 1.6 to show that,
for any/e L\ there exists a map/^: [0, oo)d χ Ω -* X such that/^ (и, ·) is a
representative of Tuf for each u, and such that, for a.e. ω,/^ζ-,ω) is Bochner
integrable on each interval [0, u) with respect to Lebesgue measure. We may put
и
S0,uf(a>) = ί/οο(ρ9ω)άΌ9 A0tUf= тф)"1^/.
о
Lemma 2.1 extends to the present situation with the same proof.
Lemma 2.9. Let {Tt, t ^0} be a strongly continuous 1-parameter semigroup of
Lx — L^-contr-actions in L\,fe Lf, and a>0. Then
//(sup \A0J\>2a)u-\\{T0f)a+№.
ο<ί< oo a
The proof may be obtained by discrete approximation from the maximal ergodic
lemma of Chacon in § 4.3. As in the proof of theorem 2.4 this yields
Theorem 2.10. Let {Tt91 ^ 0} be a strongly continuous 1-parameter semigroup of
Li — L^-contractionsinL\andfe L\, then \A0tf— T0f\ -> 0, (/ -> 0 + 0), holds
μ-a.e..
The full multiparameter generalization of theorem 2.10 (and also of Chacon's
Banach valued ergodic theorem 4.3) seems to require new ideas. However, in the
case of measure preserving transformations the maximal inequalities appearing
in the paper of Tempel'man [1972] are enough to prove:
Theorem 2.11. Let {τΜ: и е [0, oo)d} be a measurable semigroup of measure
preserving transformations in (Ω, si, μ), τ0 = identity, and X a Banach space. The
semigroup {Tu} in L\ defined by Tuf = f<> %u is strongly continuous. For allfe L\
Нт|Л0,вв/-/| = 0 μ-a.e..
£-►0

Non positive and multiparameter semigroups 249
The proof of the first assertion is similar to that of the special case treated in § 1.6.
Therefore, in view of the Banach valued version of lemma 2.1, the proof can be
completed with the help of the following inequality
const
ju( sup \A0>Beg\>a)S \\g№, (a>0),
0<ε<οο #
which is a special case of the results of Tempel'man. The averages A0ee may be
replaced by more general "regular" averages. Complete proofs and further
generalizations have been given in the Diplomarbeit of U. Wacker [1982].
Notes
1. Multiparameter semigroups. De Guzman [1975], Bruckner [1971], and Hayes and
Pauc [1970] are references concerning the differentiation of integrals.
Akcoglu and del Junco [1981] have proved a generalization of theorem 2.6 for
multiparameter additive processes: Let βά be the class of intervals / = [u, ν [ in Pd having
positive Lebesgue measure A (7). A subadditive process F= {Fj: Ie fd) for a semigroup
{Tu:ue Pd} of positive linear operators is a set function F: /аэ1 -» Fte Lx with the
following properties:
(i) Ти% = Ъ+и, (uePd,Iej?d);
η
(ii) if /l5 ...,/„ e /d, are disjoint and their union / belongs to βά, then Σ Fh ^ Ft;
(Hi) 8ир{||/Я|1/Я(/):/еЛ}<оо.
Theorem 2.12 (Akcoglu-del Junco). IfF is an additive process for a strongly continuous
semigroup {Tu:ue Pd} of positive contractions in Lu then £~dF[0ee[ converges a.e. along
each countable sequence ε -» 0.
Note that {Tu} need not be continuous at 0.
The proof uses a reduction to Terrell's theorem and to theorem 1.3.
Lin [1982] has used theorem 2.12 to derive a full multiparameter generalization of
theorem 1.3. For multiparameter local ergodic theorems in L^ see Lin [1982], Emilion
[1984d].
Akcoglu and Krengel [1981] have proved a local ergodic theorem for multiparameter
super additive processes: Let {τΜ: и е Pd} be a measurable semigroup of measure preserving
transformations, τ0 = identity, and Tuf = f°Tu.n(u)~iF[0u[ converges a.e. for all
decreasing (and other "regular") sequences и -* 0 + 0.
Emilion [1983] has a generalization of the theorem of Akcoglu - del Junco to "strongly
subadditive" processes {Fj}.
Emilion [1984b] considered the problem of continuous extension at 0 for a strongly
continuous locally bounded semigroup {Tu,ue Pd} of positive operators in L1. The
restriction to L1(C0kjD*) admits a continuous extension at 0. Sato [1984] studied this
problem (and local ergodic theorems) for null preserving semiflows {τΜ} and the induced
contractions Tu in Lv limM_+0 Tu = T0 exists and T0 = /iff μ(Ε) > 0 implies μίτ"1 Ε) > 0
for some и > 0.
Conze and Dang-Ngoc [1978] have local ergodic theorems for semigroups indexed by a
Lie group.

250 Local ergodic theorems
2. Non positive semigroups. Theorem 2.7 was obtained by Kubokawa [1975] assuming
strong convergence of Tt to the identity, (t -> 0), and independently by Kipnis [1974]
without this condition, Sato [1978] also gave a proof. Kipnis [1974] and Sato [1978 c]
obtained a representation for non positive contraction semigroups analogous to the
results which Akcoglu-Brunel gave in the discrete parameter case, see § 4.1. They showed
that the construction of Tt may break down if the contraction hypothesis is dropped.
Theorem 2.8 is due to Kubokawa-Kipnis. Akcoglu and Falkowitz [1983] showed that
the continuity at 0 is not needed. Emilion [1984c], [1984e] has a complete generalization
of the Akcoglu-del Junco theorem to the non positive case.
Concerning unresticted convergence, McGrath [1980] used Fava's dominated
estimates for non commuting Lx — L^-contractions (§ 6.1) to prove:
Theorem2.13. Let {Tk(t)\ t ^ 0}, (к = 1, ...,d), be strongly continuous semigroups of
Lx — L^-contractions in L1 of a σ-finite measure space; 7^(0) = 7. If
ί \ίΙΛ(\ο%\/Ιί\)Λ'1άμ<α^ for all/ >0
1/1>ί
then
—-!... \uQ...Ti(ti)fdti...dti
π(μ) ο ο
converges tofa.e. for и -> 0 + 0.
(Remark. The argument in the middle of p. 215 of McGrath's paper should be applied to
gfc(ft) = sup {\A(Tk, a)/-/|: 0 < a < ft} rather than to \A(Tk, ak) -f\, where ft > 0 is
so small that ||gfc(ft)||{ <e*+1.)
Theorem 2.13 goes back to Jessen-Marcinkiewicz-Zygmund [1935] for the semigroup of
translations in Ud; see also Fava [1972], de Guzman [1975]. Sato [1981a] remarked that
the continuity at 0 is not essential. For further generalizations see Yoshimoto [1982].
Akcoglu-Krengel [1979 a] have constructed a strongly continuous semigroup
{Tt, t^ 0} of unitary operators in L2, and an/0 e L2 for which ε-1 »SO>8/diverges а.е..
Thus in L2, in contrast to Ll9 the positivity of the Tt is essential for a local ergodic
theorem. Feder [1981] has related examples in Lp, (1 ^p < 2). Gaposhkin [1981] has
given necessary and sufficient conditions for the local ergodic theorem for a unitary group
{Tt, t e U} in L2 in terms of the spectral measure. Baxter and Chacon [1974] proved a local
ergodic theorem for non positive contractions {Tt, t ^ 0} in Lp, (1 <p < oo), assuming
the existence of a strictly positive h on [0, oo) χ Ω such that \f\^h(t, ·) implies
\T,f\uh(t + s,·).
3. Vector valued functions. Theorem 2.10 is a slight variant of a theorem of Hasegawa-
Sato-Tsurumi [1978]. They assume strong continuity only at t > 0, but take X reflexive.
They use only sup {|| T^: 0 < t < 1} < oo instead of || %(]„ й 1.
For further related results, see Yoshimoto [1979].

Chapter 8: Subsequences and generalized means
We now replace the usual Cesaro averages by weighted averages. A related
problem is the convergence of averages of Tkif, when kx < k2 < ... is a subsequence of
the integers. The first section treats mean convergence and the second pointwise
convergence.
§8.1 Strong convergence and mixing
1. Weighted averages for eigenvectors and flight vectors. Let Γ be a power
bounded linear operator in a Banach space X. We consider averages
00
A%x= Σ 0LVk1*x, (TV =1,2,...)
fc = 0
where the ocNk are real numbers (weights) satisfying
00
(Wl) aNk ^ 0, Σ aNk = 1 for all N.
k = 0
We also consider the family Я of all strictly increasing sequences 0
rg kx < k2 < ... of integers. If α = (ocNk) is the matrix of "canonical" weights
associated with к = (kt) by
(1.1) aNk = TV-1 if fee {fc1? fc2?..., kN}, and = 0 else,
N
then A*Nx is the 7V-th average A\x ·= TV"1 Σ Tkix along the subsequence. Many
i = l
matrices of weights have the properties
(W2) lim αΝί = 0 for all i and
oo
(W3) lim Σ |αΝti+1 — aNti\ = 0 uniformly in N.
fc-> oo i = k
For these a, (^) is an 5^-ergodic net in the sense of §2.1. (We could even
00 00
replace (Wl) by: Σ I алг* I й К for all N and lim Σ olNJc = 1.) By Eberlein's theorem
о о
A%x converges strongly if a subsequence converges weakly. Thus, this yields
many examples - but adds nothing new.
Recall that under fairly general conditions there is a splitting X — Xuds © Xfh
where Xuds is the closure of the linear span of the eigenvectors with unimodular

252 Subsequences and generalized means
eigenvalue, and Xfl the space of "flight vectors" having 0 in their weak orbit
closure. It is very simple to give necessary and sufficient conditions for the strong
convergence of Α%χ for all χ e Xuds. If χ is an eigenvector of Γ with unimodular
00
eigenvalue Λ, we have A%x = ( Σ aNkXk)x.
fc = 0
Hence, the condition
(W4) lim Σ ашХк = 0 for all λ with |A| = 1 and Я Ф 1
N-+00 fc = 0
implies the strong convergence of AaNx to a Γ-invariant limit for all xe Xuds.
Conversely, if A%x shall converge strongly to a Γ-in variant limit for all Γ and all
eigenvectors χ with unimodular eigenvalue, (W4) must hold.
[Hanson and Pledger [1969] have used the equivalent condition (W4')? that
oo
(1.2) lim Σ <*N,u+j = r1 for/^2 and y=0 /—1
]V->oo fc = 0
and
(1.3) lim Σ <*N,k = b-a (от0^а<Ь<1
N->oo {к: fcy mod 1 e[a,b)}
and irrational y,
hold. (W4') => (W4) is easy, but the converse is a bit cumbersome.]
At present no condition seems to be known which is necessary and sufficient
for the strong convergence of A%x for all flight vectors. We discuss a sufficient
condition for subsequences. Recall that Д, (F) was the lower density, D* (F) the
upper density, and D{F) the density of a set F a Z+ as defined in § 2.3. For к
= (кд put A, (k) .= D+({ku k29...}), etc.
Theorem 1.1 (Jones-Lin). Let Τ be power bounded, В = {he X*: \\h\\ ^ 1}, and
xe X. The following conditions are equivalent:
(1.4) D-lim(Tjx,h} = 0 for all he X*;
N
(1.5) lim^x = 0 forallkwithD:¥(k)>0;
N
N
(1.6) sup {TV-1 Σ |<7*'*,Л>|:ЛеД} -+0 as Ν-+αο
for all к with D# (k) > 0.
Proof (1.6) together with \\y\\ = sup {<>>, h}: he B} implies (1.5). To show that
(1.4) implies (1.6) we use proposition 2.4.9, which says that (1.4) is equivalent to
(1.7) supiTV"1 Σ |<Г'*,Л>|:ЛеД}-*0 (Ν-+αο).
i = 0

Strong convergence 253
ДДк) > 0 is equivalent to C-= sup kji < oo. For any A e В we have
N kN kN
Ν-1 Σ \<Tk'x,hy\uN-^\<Tx,h}\uCk^ ΣΚΛ,Α)|.
j=l i=l i=l
Therefore kN -* oo and (1.7) yield (1.6).
(1.5) => (1.4): If (1.4) fails there exists an ε > 0, an A e 2?, and a sequence 1 with
positive upper density and |< Tli x, A> \ > 2ε, (ι = 1, 2,...). As the union of finitely
many sequences of density 0 has density 0 we can assume Re ((Tlix, A» > ε by
passing to a subsequence of positive upper density 2a > 0 and multiplying the
original A with a complex number of modulus 1. Take any ρ e N and form the
sequence к consisting of the elements of 1 and all multiples of/?. As к has positive
lower density, (1.5) implies A\x -* 0. On the other hand, there are infinitely
many η with aln < n. If Μ is a bound for the || Γ11|, and kN is such an /w, we have
Re«^x, A» = Ν'1 Σ Re«rk% A» ^ N'1 [*kNz - M\\x\\kNlp\
i = i
taLe-M\\x\\C/p.
For large ρ we have a contradiction, α
For χ e Xfh condition (1.4) holds by the results of Jones and Lin at the end of
section 2.4, and (1.5) follows.
2. Weak convergence of Ρ χ implies strong convergence of A% x. We now derive a
convergence theorem for rather general weights imposing the condition of weak
convergence of Tlx. We first show that this condition is necessary.
Proposition 1.2. Let Τ be power bounded in a Banach space X. If'A^x converges
strongly for all keft, the sequence Tlx converges weakly to a fixed point.
Proof The limit is necessarily the same for all k. Otherwise we could form a third
sequence 1, consisting of pieces of sequences к, к' with different limits, for which
AlNx diverges. Looking at kt = /we see that the limit is Γ-invariant. Subtracting it
from χ we may assume that it is 0. If Tlx does not converge weakly to 0 there
exists he 2?, ε > 0, and kx<k2< ... with Re«rk% A» > ε for all i. Then
Re«^l^x, A» > ε, (Ν ^ 1), contradicts the strong convergence of A\x to 0. α
The following theorem was proved by Blum-Hanson [1960] for measure
preserving transformations, and by Akcoglu-Sucheston [1972] and Jones-Kuftinec
[1971] for general contractions in Hubert space. (These papers deal only with
subsequences).
Theorem 1.3. Let Τ be a contraction in a Hilbert space X.IfTlx converges weakly,
the averages A%x converge strongly to the same limit for all weights (ocNi) with

254 Subsequences and generalized means
(W5) c:=sup |aNi| <oo; limXaNi = 1; lim sup |aNi| = 0.
Ν Ν i N i
In particular, A\x converges strongly for all keft.
Proof. Necessarily, the limit is fixed, and we may assume it is 0. Since Γ is a
contraction lim || Tnx\\ exists. Hence, given an ε > 0, one can choose an integer
A:>0 such that к ^ К and j ^ 0 imply || Tkx\\ - || Tk+jx\\ < ε2, and also
| <Гкх, x> | ^ ε. We apply Lemma 2.3.3 to x' = Tkx, y' = x, and S = T\ and
obtain that
|<Гкх, x> - (Tk+jx, Tjx}\ ^ s\\x\\
and
|<Γλ+^,Γ^>|^ε(1 + ||χ||)
hold for all к ^ К and all j. This means that | i -j\ ^ К implies | <fx, Tjx} \
^ ε(1 + ||x||). Given η > 0, we have max,- |αΝι·| < η for large enough N. Clearly
00 00 00
НЕ^ГхН2^ Σ Σ \0LNi0LNJ<Px9Pxy\.
i=l i=l j=l
For fixed i, there are at most 2 К — 1 indices j with \i—j\<K. Thus, the sum of
these terms can be estimated by (2К — 1) | ocNi \ \\ χ \\2 η. The sum of the remaining
terms is bounded by ΣΣΙαΜαΝ/Ιβ(1 + 11*11) й s(l + ll*ll)c2· As ε and η were
arbitrarily small the proof is complete, α
3. Mixing and complete mixing. It follows from the spectral mixing theorem that
an endomorphism τ of a probability space is weakly mixing iff ZMim </° τ", g}
= 0 holds for all/e L2 with J/ = 0. It is even simpler to check that τ is mixing iff
/° τ" tends weakly to 0 for all/e L2 with zero integral. Thus, the results above are
characterizations of weak mixing and mixing.
It has been a problem dating back to Hopf [1937] to find an appropriate
notion of mixing for endomorphisms of infinite measure spaces. Let L? be the
subspace of Lx consisting of the functions with \fdμ = 0. We call a positive
contraction Tin Lx mixing if Tn/tends weakly to 0 for all/e L?, weakly mixing if
D-lim <Jnf g} = 0 holds for all/e L°u geL^, and completely mixing if || Tnf\\x
tends to 0 for all/e L?. An endomorphism τ (or, more generally, a null
preserving τ) is called mixing (weakly mixing,...), if the Lx -contraction corresponding
to τ (i.e., with {Tfg} = <f,g°T} for feLu geLJ is mixing (weakly
mixing,...). It is simple to check that mixing and weak mixing agree with the
classical notions for endomorphisms of finite measure spaces.
The following theorem of Krengel and Sucheston [1969 a] was (and perhaps
remains) quite surprising. [It provided a negative answer to the problem whether
ΑΓ-automorphisms of σ-finite measure spaces are mixing, and showed that, in
fact, invertible mixing measure preserving transformations of a σ-finite infinite

Strong convergence 255
space do not exist. On the other hand, it was shown in the same paper that
exact endomorphisms are mixing.]
Theorem 1.4. Let Τ be a positive contraction in Lufor which there exists no поп
zero pe L^ with Tp = p. Iff is integrable and Tnf converges weakly to some
element ofLx, then || Tnf\\1 -> 0. In particular, if Τ is mixing, then Τ is completely
mixing.
Proof Let L be a Banach limit. Define a positive linear functional у on L^ by
y(h) = L((J | Tnf\hd\iY). As (Tnf) converges weakly, this sequence is
conditionally weakly compact and hence uniformly integrable by the Vitali-Hahn-Saks
theorem. It follows that if An is a sequence of sets with An J, 0, then γ (1 At) tends to
0. Thus γ is σ-additive and there exists/? e L\ with y(h) = \phdμ. For h ^ 0 we
have
1{Τρ)Ηάμ = у(Г*А) = L{{\\Tnf\T*hdμ)) = L((J T\ Τ/\Ηάμ))
^Щ$\Tn+V\hdμ)) = y(h) = $phdμ.
Hence Tp ^ p. As Γ is a contraction, Jp = /?. The assumption implies/? = 0. In
particular, 0 = y(H) = lim|| Tnf\\v α
Remarks. Only the conditional weak compactness of (Tnf) was used in the
proof. If Γ is an arbitrary contraction in Lx and | T\ its modulus, the argument
shows | T\p = p. Hence, if | T\ has no positive non zero fixed points, and (Tnf) is
conditionally weakly compact, again || Tnf\\1 -► 0 follows.
Notes
Cohen [1940] considered mean ergodic theorems with weights now covered by Eberlein's
theorem; see also Kurtz and Tucker [1968]. Hanson and Pledger [1969] showed that
(\¥4') is necessary and sufficient for the strong convergence of AaNx for all isometries in
Hubert space using a reduction to the unitary case in which this follows from the spectral
theorem. They also showed for the isometries in L2 induced by endomorphisms of a
probability space that A%x converges strongly for all flight vectors if the weights satisfy
(W6) Σα^->0 for all FaZ+ with D(F) = 0.
keF
It seems to have been overlooked that this contains the special case of measure preserving
transformations of Jones' [1971] result, that the weak convergence of Tnix along a
subsequence of density 1 implies (1.5). This implication follows from the easy half of
Proposition 1.5. If (ocNi) is the matrix of canonical weights for k, the condition (W6) is
equivalent to Д, (k) > 0.
Jones [1972] proved
Theorem 1.6. If Τ is a contraction in a uniformly convex space X the following are
equivalent:

256 Subsequences and generalized means
(1.8) ЗЕ = ЗЕЛ;
(1.9) Given ιφΟ there exists βχ>0 with Д,({«: \\Tnx-x\\< βχ}) = 0;
(1.10) Given x, there exists F with D(F) = 1 and lim inf || Tnx - x\\ ^ ||x||.
neF
For related material, see Jones [1973], Nagel [1974], Patil [1977], Blum-Mizel [1971].
Akcoglu, Huneke and Rost [1974] showed that theorem 1.3 does not extend to arbitrary
Banach spaces. Their counterexample is a contraction in C(K) for compact K. It was
further analysed by Sine [1974].
Akcoglu and Sucheston [1972], [1975] proved the equivalence of
(A) w-\imTnx = y
and
(B) AaNx -► у for all weights with (W5)
for contractions in Lu in L2, and for positive contractions in Lp, (1 <p < oo).
Weights with (W5) were first used by Fong-Sucheston [1974] for these problems. Mrs.
Bellow [1975], [1976] gave a proof for Lp, (l<p< oo), of (А) о (В) avoiding the dilation
argument of Akcoglu-Sucheston. Hiai [1978], Iwanik [1979], and Blum-Eisenberg
[1979] have generalizations for certain groups of operators. Mrs. Millet [1980] proved
(А) о (В) for positive power bounded Tin Lu for certain non positive operators in Ll9
and for positive contractions in Orlicz spaces having a uniformly convex dual; see also
Sato [1975 b], [1976]. These papers contain more references.
Mixing for Markov operators was studied by Lin [1971].
Lin [1972a] extended theorem 1.4 to power bounded rand showed that Τ is mixing iff
Γ χ Γ is mixing. (The terminology in these papers is different).
Aaronson, Lin and Weiss [1979] showed that the following conditions are equivalent to
weak mixing for a positive contraction in L^
(1.11) Tis ergodic (i.e., ΜΝ/||! -> 0 for all/e L?), and T* has no unimodular
eigenvalues φΐ;
(1.12) For every ergodic positive contraction S in Lx with finite invariant measure
Τ χ S is ergodic.
Lin [1981] has characterized weak mixing by
(1.13) Forevery/ze L^ there is a sequence^· of density 1 such that all w* -cluster points
of {Γ*Πί/ζ} are constants.
00
Foguel [1973] studied the convergence of Rnx, where R = J φ(ή Ttdt is an "average"
о
over a semigroup {Tt, t ^ 0}.
Friedman [1983] calls an endomorphism τ of a probability space (Ω, л/, μ) mixing
alongkif μ(Α ητ~ΗίΒ) -► μ(Α) μ(Β) holds for all A, Be я/ and studies the Blum-Hanson
theorem along such sequences.

Pointwise convergence 257
§ 8.2 Pointwise convergence
1. Weighted and Abelian averages. Some ergodic theorems and maximal inequal-
00 00
ities deal with Abelian averages like λ J e~Xs TJds or (1 - λ) Σ Лк Tkf We shall
0 fc = 0
see in the Notes that the investigation of such averages can make sense.
Frequently, however, ergodic theorems about Abelian averages do not deserve a
separate proof. As Cesaro convergence implies Abel convergence, the ordinary
ergodic theorems imply Abelian·ergodic theorems as corollaries. We derive a
variant of this classical result which should suffice for most ergodic theoretic
applications, including ratio theorems.
00
Consider non negative weights aXk with Σ αλл = 1 for all λ e Л, Л an arbitrary
fc = 0
И 00
directed set. For a sequence (fk) put Fn — Σ Λ» and Ρχ = Σ ал/сЛ> whenever
fc=0 fc = 0
this sum is well defined. Similarly, (gk) defines Gn and Gax.
Theorem 2.1. Assume the weights (сслк) satisfy
(W7) αλ* ^ ал,к+1 for all А, к and
(W8) lim {<χλΛ - aAtk+1) = 0 for all к.
л
00
If(fk) is a sequence of vectors in a Banach space which satisfies Σ 11 ^xkfk 11 < °ofor
k = 0
all λ, and if(gk) is a sequence of non negative numbers, not all 0, then FJGn -> χ
implies F£\G\ -> x.
Proof The assumptions permit the use of the Abel transformation
(2.1) /5-= Σ Kk-^,k+1)Fk, GJ= Σ Kk-aA,fc+1)Gfc.
fc=0 fc = 0
First assume Gn -► 00. Then (W8) and (2.1) imply GJ -► 00. Put £k = Fk/C?k - x.
Then
00
(2.2) \\F?-xGl\\u Σ («яд-«яд+1)Н^||^.
fc = 0
For ε > 0 there exists AT with || /?k|| < ε for k^ K. The sum of the first AT terms on
the righthand side of (2.2) tends to 0 after division by G\, and
Σ Κ*-αΑ.Λ+1)|ΐΑΐΐσΛ/σ2ί8 us.
k = K
Now assume that the limit G of (Gn) is finite. Then Fn tends to a limit F. For
ε > 0 there exists L > 0 with ||/£ — F|| < ε for k^L. Hence

258 Subsequences and generalized means
\\Ff-F\\u Σ (^,k-^,k+1)sup\\Fj\\ + s,
ί = 0 j
and (W8) shows F£ -* F. Similarly, Gl^G. □
Examples. 1) The weights αλΙί = (1 - λ)λ\ (λ e Λ = ]0,1 [) satisfy (W7) and
(W8) for λ -+ 1. If Τ is a contraction in some Lp, then Σι I TkfXk | < oo for all λ
00
holds a.e.. If An{T)f converges a.e., (1-Я) Σ Як Гк/converges a.e. for λ -* 1.
fc = 0
2) Let w0 ^ wx ^ ... be a decreasing sequence of positive numbers with diver-
gent sum. Then the weights with aNk = wk/Y,wkfork S N, = 0 for к > TV, satisfy
(W7) and (W8). °
The next lemma will show that maximal inequalities and dominated ergodic
theorems extend to many weighted averages:
Lemma 2.2. If /0, ...,/„ are real numbers, g0 > 0, g1?..., gn ^ 0 and w0 ^ wx
^ ... ^ wn > О, /Лея
η τ> m„ /ο + -·-+/* > m„ wo/o + -- + wfc/fc n
(2.3) max 2; max =·· ρη
ο^η£0 + ··· +£* o^fc^n w0£0 + ... + wk£k
Proof. Let /c be the first index for which the quotient on the right hand side equals
Qn. Then
(2.4) wjfj + ... + wkfk ^ ρη(ν% + ... + wkgk)
for ally with 0 rgy rg Λ;. Find ξ0,..., <i;k successively by solving the equations
i
w,- Σ £i = 1 f°r7 = 0,..., fc. The monotonicity of the w,- implies £,· ^ 0. Multiply
i = 0
the inequalities in (2.4) by £,· and sum over j=09...9k. Then (/0 + ... +/k)
^Qn(go + --+gk) follows, α
The lemma, applied with gk = 1 for all k, shows that dominated ergodic theorems
for Cesaro averages An(^/immediately yield dominated ergodic theorems for
Abelian averages as corollaries. If Γ is a positive contraction in Lx and / is
integrable, the lemma implies
(2.5) E'-.= { sup (1-Я) Σ Afcrk/>0}d{sup^w(r)/>0}.
0<A<1 fc = 0 n^l
As {/> 0} is contained in E\ (2.5) and Hopf's maximal ergodic theorem yield the
Abelian maximal ergodic theorem
J/ty^O.
E'

Pointwise convergence 259
2. Good and bad subsequences. Recall that the Blum-Hanson theorem asserts the
η
L2-norm-convergence of A*f=A*(T)f-.= n~1 Σ /°τ1ίί f°r all кеЯ and all
fe L2, when τ is a mixing endomorphism of a probability space (Ω, «я/, μ). By a
simple approximation argument, Lp-norm-convergence holds for all fe Lp,
(1 ^p < oo). The problem of finding sufficient conditions for a.e.-convergence
has generated great interest in spite of the present lack of applications. Friedman
and Ornstein [1972] showed by example that the assumptions of the Blum-
Hanson theorem are not sufficient. Then Friedman [1970] asked whether a.e.-
convergence holds for ΑΓ-automorphisms, and also whether a.e.-convergence
holds for mixing automorphisms when к has positive density. Krengel [1971]
answered the first of these questions negatively even for Bernoulli shifts by
showing that there exists a universally bad sequence к е Я, in the sense that for every
aperiodic automorphism τ of a probability space there exists an/e Lx for which
A\f diverges a.e.; see Theorem 6.4.6. The second question of Friedman is still
open, but Conze [1973] showed that there exists а к with ДДк) > О, a weakly
mixing τ, and an/e Lx for which A\/does not converge a.e.. He also showed that
the combination of the conditions proposed in the two questions is sufficient. To
prove this we begin with some general considerations.
Define C(k) as the infimum of all С > 0 with the following property: For any
automorphism τ of any probability space (Ω, «я/, μ), any fe Lu and any λ > 0
(2.6) μ(*τφΑΪ\/\>λ)Ζγ\\/\\ί.
If no finite С with this property exists, put C(k) = oo.
Note that ДДк) > 0 is equivalent to c*= sup(/cf + l)/i < oo. Using
A\ l/l = n-1 £ l/l· τ" й ^- · 1-1— Σ l/l ° τ1 й cAkn+11/|
i = l П Kn + 1 i = o
and the maximal ergodic inequality, we obtain C(k) ^ c. Thus D^ (к) > О implies
C(k)<oo.
An automorphism τ is said to have Lebesgue spectrum if the subspace L\ of L2
orthogonal to the constant function H is the closure of the linear span of a family
{fin: i e /, η e 2) of orthogonal vectors with/ w+1 =fUn ° τ for all i e /, η e Z. (/is
an arbitrary index set). It is well known that ΑΓ-automorphisms have Lebesgue
spectrum.
Theorem 2.3 (Conze). Let τ be an automorphism with Lebesgue spectrum. Then
A\f converges a.e. for all fe Li and alike Si with Д<(к) > О.
Proof. The theorem of Rajchman (see e.g. Chung [1968: 97]) asserts that the
sequence of Cesaro averages of any bounded sequence of orthogonal vectors in
L\ convergences a.e. to 0. Thus, A\^fitn convergences a.e. to 0 for allfUn. Clearly,

260 Subsequences and generalized means
A^l = 1. Thus, ^/converges a.e. for all/in a subset of Lx which is dense in Lt.
The estimate (2.6) implies that the set of all/e Lx for which ^/converges a.e. is
closed (Theorem 1.7.2). α
Obviously, periodic sequences к are universally good in the sense that
^/converges a.e. for all automorphisms τ of an arbitrary probability space and all
integrable/. The first non trivial class of universally good sequences к has been
described by Brunel-Keane [1969] as follows:
Let σ be a homeomorphism of a compact metric space Σ with metric ρ such
that all powers ση are equicontinuous, and assume that there exists ze Σ with
dense orbit {anz:ne N}. Then there exists a unique (and hence ergodic), σ-
invariant probability measure ν on &, the σ-algebra of Borel sets, see § 1.2. Each
non empty open set has positive v-measure. A sequence к е Я is called uniform if
there exists such (Σ, ^, ν, σ), and a set Υe Μ with v(5 7) = 0 and ν(Υ) > 0, and a
pointy g Σ with ki = i-th entry time of the orbit of у into Y. (Formally:
kt = inf{l^0: Σ Utfy) = Ο)· S Υ is the boundary of Y.
Theorem 2.4 (Brunel-Keane). Uniform sequences are universally good.
Proof Let ε > 0 be given. For small δ > 0 and very small <5' = δ'(δ) > 0 the
open sets Γ = (χ6 Υ: ρ (χ, δ Υ) > δ} and ^ = {χ β Σ: ρ (χ, у) < δ'} satisfy
μ(Υ\Υ')< ε and
(2.7) 17,(σηχ) ^ ly^jO for allxeW,n^ 0.
Consider the Cartesian product (Ω', .я/', μ') = (Ω, .я/, μ) χ (Σ, ^, ν) and the
product automorphism τ' = τ χ σ defined by τ'(ω, χ) = (τω, σχ). For given
/eL^/i) the functions g, g' on Ω' defined by #(ω, x):=f(a>)lY(x) and
#'(ω> лг):=/(ш)1Г/(л;) are μ'-integrable. Thus, g' = limAn(T')g' exists μ'-a.e..
Using the uniform integrability of the An(x')g' and the ergodicity of σ we find
J £'<//ι'= Ιίιη/Γ^Σ JJ Κτκω)\Τ(σκχ)μ(άω)ν(άχ)
QxW n->oo fc = 0 βχ W
= J/rf/iUm J л-1 "Σ 1Г(***М</*) = v(W) ν(Γ) J/V/μ.
Ж fc = 0
Now (2.7) implies that §(ω) --= lim mf(An(x')g) (ω, у) satisfies g'(u>, x) й §(ω)
for ω e Ω, χ e W. Hence
JS<//i = v(W0_1 J J §(ω)άμ,^ν(ΐν)-1 J J g'<//i'
βχ w βχ W
= ν(Γ)|/φ^(ν(7)-ε)ί/^.
A symmetric argument shows that S((o)-—lim sup(A„(x')g) (ω,y) satisfies
J Si/μ й (ν(Υ) + ε) \/άμ. Together these estimates yield \{S- §)άμ £2&\\f\\v

Pointwise convergence 261
As ε > 0 was arbitrary, S = S μ-a.e.. Thus
(2.8) (An(z')g) (ω,у) = и"1 "Σ /(τ'ω)ly(A)
i = 0
= "_1 Σ /(τ*ω)
{i:kiun-l}
converges μ-a.e.. Setting/ = 1, we see that n~x card {/: fcf rg и — 1} converges to
v(Y)9 and then (2.8) yields the convergence μ-a.e. of A*f. α
Remark. If τ is weakly mixing, τ' is ergodic and the limit is constant.
3. Random ergodic theorems. While the ergodic theorems considered so far deal
with iterates of a fixed operator or with semigroups, random ergodic theorems
deal with sequences of operators for which the operator applied at time к
depends on the state in which a stochastic process is at time k. Again, applications
seem scarce, which allows us to restrict attention to the simplest case.
Let σ be an automorphism of a probability space (Σ, M9 v), and let (Ω, s/9 μ) be
another probability space. Assume that for each xe Σ there is an endomorphism
τ(χ) of Ω, such that the skew product τ' defined in the product measure space
(Ω', st* 9 μ') = (Ω, sf9 μ) χ (Σ, Зй9 ν) by τ'(ω, χ) --= (τ (χ) ω, σχ) is measurable. For
any xe Σ put Θ0(χ) = identity in Ω, and Θη+1 (χ) = τ(σηχ) ° Θη{χ).
Theorem 2.5 (Random ergodic theorem for endomorphisms). For anyfe Lx (μ)
n-l
there exists a set Nfe& with v(Nf) = 0 such that и-1 Σ /° 0fW converges
μ-a.e. for χ еЩ. i = 0
Proof. The assumptions imply that τ' is an endomorphism of Ω'. Put g{a>9 x)
= /(ω). Induction yields (τ')*(ω, χ) = (0^(х)ш, σ'χ). Hence g((Tfy(a>9x))
= (/° 0f(x)) (ω). BirkhofFs theorem implies the convergence μ'-a.e. of An{x')g.
Let N' be the μ'-nullset on which there is divergence. Then
Nf--= {xe Σ: μ({ω: (ω, x)e Ν'} > 0)} is a v-nullset. For xeNcf9
n-l n-l
л-1 Σ #((τ')4ω? *)) = n~l Σ (/° ®г(*)) (ω) converges μ-a.e.. α
ί = 0 i = 0
Example. Let σ be the shift in Σ = {χ = (x0, x1?...): xt e {1, 2}}, where the
coordinates are independent under ν with v({x: xt — 1}) = 1/2. Let τ be a fixed
automorphism of (Ω, λ/, μ) and τ (χ) = τ*°. Then 0f(x) = xki with fcf = x0 + ...
+ xi_1. We obtain convergence μ-a.e. of A*f for almost all subsequences к
= k(*).

262 Subsequences and generalized means
Notes
The books of Hardy [1949] and Zeller-Beekmann [1970] are well known references for
the theory of summability. It is interesting to note that one can also deduce Cesaro
convergence from Abelian ergodic theorems using the following Tauberian theorem of
Hardy and Littlewood (cf. Hardy [1949:155]):
00
Theorem 2.6. If(an) is a bounded or поп negative sequence, and (1-Я) Σ kkak tends to χ
for λ -> 1 — 0, then η 1 Σ ak tends to x.
fc = 0
Rota [1963] remarked that the proof of the Abelian maximal inequality is somewhat
simpler than the proof of Hopfs inequality. But the Tauberian theorem is not so easy, and
thus a direct proof of Cesaro convergence seems preferable.
Several authors have produced counterexamples to show that certain conditions cannot
be weakened in Abelian ergodic theorems. By theorem 2.6, these examples are superfluous
when the corresponding Cesaro type examples exist.
Krengel [1967] used a variant of theorem 2.1 and lemma 2.2 to point out that weighted
ratio ergodic theorems and maximal inequalities for decreasing weights do not require a
separate proof. The argument can be modified for continuous parameter semigroups. If
00
weights of the form aAs = J ρλ (и) du with integrable ρλ ^ 0 are given, one can perform a
transformation s
00 00 S
ί ocXsf(s)ds = J qx(s) \f(u)duds.
0 0 0
E.g. for aXs = Ae~As one takes qx{u) = λ2 e~ku, and finds: If e~Xsf(s) is integrable for all λ
t t
withO < λ < oo, andg ^ 0 has positive integral, then lim ($f(s)ds/$g(s)ds) = χ implies
oo oo ί-oo О О
lim (J e~ksf(s)ds/ J e~Xsg{s)ds) = x. This remark yields ergodic theorems for resolv-
Λ-0 0 0
ents. Similarly, Abelian local ergodic theorems may be deduced from ordinary local
ergodic theorems. Details have been given by Sato [1980], [1981c]. (Thesepapers contain
more references.)
The following Tauberian theorem is a special case of the results of Karamata [1931]; see
Widder [1946:192].
00
Theorem 2.7. If F(t) is поп decreasing, and such that the integral rk = J e~Xt dF(t) con-
о
verges for λ > 0, then Xrx -► χ, (λ -► 0 + 0), implies t x F(t) -> x, (t -» oo), and Xrx -» x,
(λ -► oo), implies t~1F(t) -> x, (t -+0 + 0).
Feyel has systematically employed this theorem to deduce local and global ergodic
theorems for continuous parameter semigroups. He proves his results even for pseudo-
resolvents; see § 2.1 (Notes). He calls a pseudo-resolvent for which the λ Vx are positive
contractions of L± proper if Vxco = oo, (i.e., 0 <Ξ/Β | oo implies VJn \ oo). The following
maximal ergodic theorem of Feyel [1977] does not follow from Hopfs maximal ergodic
theorem even for resolvents:
Theorem 2.8. Let (λ Vk)k > 0 be a pseudo-resolvent for which the λ Vx are positive contractions
in Lu and which is proper. Forfe Ьг andD a R4" put Ε = {supAeD λ Vxf> 0}, where sup is
the "essential" supremum in Lv Then \Εί^μ ^ 0.

Pointwise convergence 263
The subsequent papers of Feyel [1977a], [1977b], [1979], [1982], [1982a] contain
pseudo-resolvent forms of the Chacon-Ornstein theorem, the Dunford-Schwartz
theorem, Brunei's lemma, and of local ergodic theorems for superadditive processes. The
Abelian analogue of a superadditive process is a superabelian process, i.e., a family of
measurable functions (ηλ)λ>0 with \\u^\\t^ Μ/λ for some M<co, and ux^uQ
+ (ρ — λ) Veux, (0 < λ 5Ξ ρ < oo). Using an extension of the notion of the modulus for
pseudo-resolvents (c.f. Kipnis [1974], Feyel [1978]), also non positive pseudo-resolvents
have been investigated by Feyel [1984].
If (AVA) is bounded and continuous at oo, and T0f= \imx^OQXVxf, the Hille-Yosida
theorem implies that there exists a strongly continuous semigroup on T0LU such that
(λ νλ) is the resolvent of this semigroup when restricted to T0 Lx. As the Dunford-Schwartz
theorem, the Chacon-Ornstein theorem, etc. hold with the same proofs for the subspace,
they yield in turn ergodic theorems for pseudo-resolvents. Feyel [1979] showed that
proper pseudo-resolvents of positive contractions in Lx are continuous at oo.
Feyel [1982 a] does not need the assumption that the pseudo-resolvents are proper for
his results on superabelian processes. Also, the class of superabelian processes is strictly
larger than the class of Laplace transforms of superadditive processes.
Tauberian theorems for sequences of ratios do not seem to be known. Thus, at present,
it is not clear if the Abelian form of the Chacon-Ornstein theorem, which is simpler to
prove, could be used to deduce the Chacon-Ornstein theorem.
Baxter [1964], [1965] has proved weighted ergodic theorems with "recurrent" weights
(μΌ*£ο given recursively by w0 = 1 and wn = p^wn^ +/?2w„_2 + ... + #,w0, where
(Pk)k^i is a probability distribution on N. Chacon [1964] has shown that the a.e.-
n η
convergence of ratios Σ wi Tlfl Σ wi Tlg for positive contractions Γ in Lx and/, g e L\
i = 0 i = 0
can also be deduced from the Chacon-Ornstein theorem. Berk [1968] has proved
continuous parameter ergodic theorems of this type.
Irmisch [1980] has studied (C, a)-averages
in + a\-1 "/«-/ + a -1\ .,.
4Λ/,α)'= Σ . )Tlf.
\ η J i = 0\ n-i J
He proved the а.е.-convergence of these averages, when Γ is a positive contraction in Lp
with 1</?<οο,0<α£Ξ1, and cap > 1. He also gave a counterexample with ap = 1.
Akcoglu and del Junco [1975] showed that Aan (τ)/ need not converge a.e. for weights
(a**) with (W4).
In the situations above, the convergence of Cesaro averages was known. A different
point of view is that weighted averages should be used in cases where Cesaro averages fail
to converge: Krengel [1967] showed that for any positive contraction Tin Lx there exists
oo η η
Wo^wi^ ··· wrtn Σ wi = °°> sucn that Σ и^-Г4//Σ wi converges a.e. for all
i = 0 i = 0 i = 0
fe L^Grillenberger and Krengel [1976] showed that for any such Τ there exists a
sequence n1<n2< ... such that An.(T)f converges a.e. for all/. On the other hand, it is
shown in the same paper that there does not exist any matrix summation method
compatible with the Cesaro method which enforces a.e. convergence of the averages of Tnffor all
positive contractions Tin Lt.
Assani [1985 a] constructed weighted averages of L1 — L^-contractions for which
pointwise convergence holds, but which are not of strong type (p, p).
Good and bad sequences. Conze [1973] proved that a subsequence к is bad universal iff
C(k) = oo holds, and used this result to give simple examples. Mrs. Bellow [1982], [1984]

264 Subsequences and generalized means
proved C(k) = oo for sequences к for which there exists a λ > 1 with ki+i/ki > λ for all i,
and she has constructed other bad universal sequences. Bellow and Losert [1984] showed
the existence of good universal sequences к with D (k) = 0.
The theorem of Brunel-Keane has been extended to more general operators by Sato,
Olsen, and others. The paper of Baxter-Olsen [1983] contains references, gives a more
systematic approach, and proposes new classes of universally good sequences. Here the
original special case has been presented for reasons of simplicity. Ryll-Nardzewski [1975]
showed that, for uniform k, the sequence (a,·) with a,- = 1 if there exists an i with k{ = j, and
oij = 0 else, is a bounded Besicovich sequence. (A bounded sequence (а,·) с С is called
bounded Besicovich sequence if, for any ε > 0, there exists a trigonometric polynomial
η
wsU) = Σ7s exP (i@sJ) witn hm sup и-1 Σ Ια,- — we(j)\ <Ξ ε.) Ergodic theorems for uni-
form sequences are special cases of ergodic theorems for averages η'1 Σ XjTjf.
i = 0
B. Schmitt [1972] showed that there is uniform convergence in the Brunel-Keane theorem
if τ is weakly mixing and strictly ergodic in a compact Ω,/e C(Q).
Blum and Reich [1977] gave another proper extension of the notion of uniform
sequences by introducing "/^-sequences". J. Reich [1977] showed convergence a.e. of A\f
for/in a dense subset of L2, "saturated" sequences к and weakly mixing τ. He used this to
prove a random ergodic theorem in which the exceptional nullset of sequences к does not
depend on τ. Related results appear in Blum-Reich [1977a].
Brunel-Keane [1969] showed that a.e.-convergence can be enforced in the Blum-
Hanson theorem by using suitable decreasing weights.
Bellow and Losert [1985] have written a long partly expository article on this circle of
problems.
Random ergodic theorems. The first random ergodic theorems were proved by Pitt
[1942], Ulam and von Neumann [1945], and Kakutani [1951]. The idea of applying a
"non-random" ergodic theorem to a product space is a recipe which has furnished
numerous random ergodic theorems. Jacobs [1962/63] has summarized and extended most of
the random ergodic theorems appearing before 1962 by proving
Theorem 2.9. Let σ be an endomorphism of a probability space (Σ, &, v), and{Tx, xe Σ} a
family of contractions in Lx of a σ-finite space (Ω, д/, μ). For any fe Lx let the
map φ/, χ -► TxfofI into L1 be strongly measurable (i.e., ν ° φ J1 has separable support).
Let Ρο,Ρι,... on Ω χ Σ be stf ® ^-measurable and admissible (i.e., \g\ ^рк(ах, ·) implies
\Txg\ ^pk+1(x,- )for k^O andv-almost allx). Then there exists, for anyfe Llfa v-nullset
N such that for xe Nc the ratios
η η
Σ TxTax...Tak~ixfi Σ Pk(x,-)
k=0 fc=0
00
converge μ-a.e. to a finite limit on {ω: Σ Ръ(х, ω) > 0}.
fc = 0
We refer to Yoshimoto [1976], [1977], Kin [1972], Tsurumi [1972], Wos [1982], De-
lasnerie [1977] and J.Reich [1981] for further random ergodic theorems
(multiparameter, multidimensional, local, etc.) and for references.
Revesz [1960/61] has random mean and pointwise ergodic theorems for averages of
functions /о тк ° тк_!... ° τΐ5 where the τ·χ are chosen independently, but not necessarily
with identical distributions.
Saleski [1980] has a random mean ergodic theorem with the exceptional nullset N

Pointwise convergence 265
independent of τ and/: Let Sn be an integer valued random walk on (Σ, &, v): There exists
N with v(N) = 0 such that, for all weakly mixing τ and all fe Lp, (1 <Ξ ρ < oo), xe Nc
η
implies the Lp-convergence of и-1 Σ /nSiW.
k = l
Ergodic theorems for sub-orbits and super-orbits. In the random ergodic theorem, the
transformation τ(σηχ) applied at time η is independent of ω. Recently, Kieffer-Dunham
[1983], and Kieffer-Rahe [1982] have proved ergodic theorems, in which the
subsequence, along which averages are taken, may depend on ω. The setting is the following: τ
and ξ are measurable transformations of a probability space (Ω, я/, μ), and τ is invertible
with measurable inverse. For each ωβΩ, ξω belongs to the bilateral orbit Θτ(ω)
= {τ*ω: ie Ζ}.
Thus, {ω, ξω, ξ2 ω,...} is a subset of Θτ(ω). Write ω' > ω if there is some η ^ 1 with ω'
= τηω.
Theorem 2.10 (Kieffer-Rahe). If τ is measure preserving and the sets {ω' e Ω: ω' < ω and
ξω' ^ ω} and {ω' e Ω: ω < ω' and ξω' 5Ξ ω} are finite for μ-almost all ω, then μ is
asymptotically mean stationary for ξ. (In other words: An(£)f converges μ-a.e./or hounded
measurable f)
There even exists A with μ {A) > 0 such that μΑ with μΑ (В) = μ (A)"χ μ (Α η Β) is invariant
under ξ and asymptotically dominates μ with respect to ξ.
There is a similar theorem when ξ is measure preserving:
Theorem 2.11 (Kieffer-Rahe). If ξ is measure preserving and aperiodic, and the map L with
ξω = tL(u>) ω satisfies J | L \ άμ < oo, then there exists a τ-invariant ν $> μοη (Ω, stf). In
particular, An(x)f converges μ-a.e. for hounded measurable f
Kieffer and Dunham consider a stationary sequence (Xt) with values in a space 2ί, another
space 93, and a function F: 2ί χ 93 -> 93. A 93-valued process (St) is generated recursively
by S0 = b* (fixed g 93), Si+i = F(Xh St). They give a sufficient condition (trivially satisfied
for finite 93) for the sequence {№, ^)ί=01) } to be asymptotically mean stationary.
There are applications to information theory and queueing theory.

Chapter 9: Special topics
This chapter deals with a number of results which require special methods and do
not fit well into the previous chapters. The first section treats ergodic theorems in
von Neumann algebras, the second ergodic theorems for entropy and
information and the third ergodic theorems for nonlinear operators. In the last section
we report on miscellaneous related topics.
§9.1 Ergodic theorems in von Neumann algebras
This section deals with questions of convergence of averages
An(T)x = n-1(T° + T1 + ... + Tn~l)x
when Γ is a *-automorphism of a von Neumann algebra, and with similar
problems for more general operators and semigroups. Here we cannot develop all
the tools from the theory of von Neumann algebras required for the proofs. But
all notions will be defined, and the meaning of the statements should become
clear also to non-specialists. References will be given for the auxiliary results,
except for some basic facts which can be found in the early parts of the
monographs of Takesaki [1979], Pedersen [1979], Sakai [1971] or Dixmier [1957].
The motivation for the results in this section arose in quantum statistical
mechanics and quantum field theory. An introduction is given in the books of Ruel-
le [1969] and Bratteli-Robinson [1979]. Roughly speaking, states in a "classical"
system may be described mathematically as linear functionals μ (Radon
measures) on the space C(K) of continuous functions on a compact hyper-Stonean
space K. Homeomorphisms of К induce *-automorphisms of the commutative
von Neumann algebra C(K). States of quantum systems may be described by
positive normed functionals ρ on a von Neumann algebra 9Ϊ. Invariant states ρ
under *-automorphisms Γ may therefore be regarded as generalizations of
invariant measures to von Neumann algebras which need not be commutative. Their
study has therefore been called non commutative ergodic theory. The words
"non commutative" refer to the multiplication in 9Ϊ; this must be distinguished
from ergodic theorems for non commutative semigroups.
1. Basic notions. A complex Banach space 9Ϊ with norm || · || is called a Banach
algebra if it is an algebra satisfying \\xy\\ S 11*1111^ II· A mapping χ -> χ* of 91
into itself is called involution if (л:*)* = χ, (Λχ)* = Лх*, (Я е С), (х + у)* = х*

268 Special topics
+ y*9 and (xy)* = y*x*. A Banach algebra with involution is called C*-algebra
if" || jc* jc || = ||x||2. 2Ϊ is called unit al if it contains a multiplicative unit 1. It seems
unnecessary and unusual to distinguish 1 e 9Ϊ from the real number 1 in the
notation. Clearly, 1* = 1,||1|| = 1.
A mapping Γ of a C*-algebra 9Ϊ into a C*-algebra (£ is called a *-homomorph-
ismif Γ (Ax) = λΤχ, Τ (χ + у) = Τχ + Ту, Т{ху) = (7х) (7>), and Гх* = (7х)*
holds for all χ, >>, Я. Then || Tx || ^ || χ ||, with equality for injective Γ; see Pedersen
[1979:16]. The notions*-isomorphism, *-automorphism, etc. are now obvious.
As any "abstract" C*-algebra is *-isomorphic to a subalgebra 9Ϊ of the space
!£ (£) of bounded linear operators on a complex Hubert space £, we may assume
throughout 9Ϊ с if (£>). The involution χ -* χ* in 91 is the passage to the adjoint
operator. A subset Μ of a C*-algebra 9Ϊ is called self-adjoint if χ e Μ implies
x* g Μ. χ is called self-adjoint if x* = x. %sa denotes the set of self-adjoint
elements of 9Ϊ. Any χ g 9Ϊ has a unique representation χ = у + iz with y9ze 9lsa.
χ g 9Ϊ is called positive (in symbols χ ^ 0) if there exists у e 9Ϊ with χ = >>*>>.
The set 2Ϊ + of positive elements of 9Ϊ is a convex cone. xx ^ x2 iff x2 — *i ^ 0
defines a partial order in 9Ϊ. Any χ e 9Ϊ5Λ has a unique representation χ = у — ζ
with у ζ = 0 and >>, ζ ^ 0. A linear functional φ: 9Ϊ -* С is called positive if χ ^ 0
implies φ (χ) ^ 0.
A C*-subalgebra of the algebra if (§) is called а шл Neumann algebra if it
contains the identity \ — I and is closed in the weak operator topology (or -
equivalently - in the strong operator topology).
A linear functional φ on a von Neumann algebra 9Ϊ is called normal if for each
norm bounded monotone increasing net {xa} in SHsa with limit χ (in the strong
operator topology) the net φ (χα) converges to φ (χ), φ is normal iff the restriction
of φ to the unit sphere 9lx of 9Ϊ is continuous in the weak operator topology. The
set 91^ of normal linear functional is a Banach space and 9Ϊ is the dual (2Ϊ*)*.
Therefore 2Ϊ* is called the predual of 2Ϊ; see Pedersen [1979: 55].
We remark that, by a theorem of Haagerup [1975], a positive linear functional
on 9Ϊ is normal iff φ (Σ хд = Σ Ψ (хд holds for any set {xj с 9Ϊ + for which Σ χ( is
defined.
Positive linear functionals on 9Ϊ having norm 1 are called states. A state φ is
called invariant under a *endomorphism Γ if φ(Τχ) — φ (χ) holds for all x.
To gain some first orientation let us consider the C*-algebra L^in) of a
probability space (Ω, sf, μ)./* is the function taking the complex conjugate values of/.
L^ is *-isomorphic to a von Neumann algebra 9Ϊ с if (L2) by/-* Xy with Xyg
= /· g. If we identify L^ with 9Ϊ = {xf: fe L^}, we see that Lx is the predual 91^.
he Lx corresponds to the functional \ph(f) = \/Ηάμ. Any automorphism τ of
(Ω, sf, μ) defines a *-automorphism Τ of L^ by 7/ = /° τ. The functional t/^
corresponding to the constant function 11 is Γ-invariant. For suitable τ and/the
sequence n~l (f+f° τ + ... +f° τ""1) does not converge in L^-norm. Thus, the
existence of normal invariant states does not imply Twrra-convergence of An{T)x

Von Neumann algebras 269
for χ β 9Ϊ. However, in this example we do have norm-convergence of Cesaro
averages for the operator induced in the predual and therefore w*-convergence in
L^. The extension of this to the non commutative case has been initiated by
Kovacs-Sziics [1966]. Birkhoffs theorem in this example asserts a.e.-
convergence which, by Egoroff, is equivalent to the existence of sets Ε with μ(Ε)
arbitrarily close to 1 for which there is L^-norm-convergence on E. In this form,
Birkhoffs theorem has been extended to the non commutative case by the
remarkable work of C. Lance [1976].
Before we proceed to these results let us remark that the existence of a ^-invariant
normal state is not enough for w*-convergence unless it has an additional
property: Assume above that μ is σ-finite, but infinite and that there exists a τ-
invariant set F with μ (F) — 1 such that the restrictions of τ to Fand to Fc both are
ergodic. Then the functional corresponding to 1F is a Γ-invariant normal state,
but we do not have Lx-convergence or w*-convergence on Fc.
A positive normal state φ is called faithful if χ ^ 0, φ (χ) = 0 implies χ = 0, i.e.,
φ is strictly positive on 31+ \ {0}. Above, \ph is faithful iff A is strictly positive а. е..
This is what we need. A family Φ = {ψι · ie 1} of positive normal states will be
called faithful if χ ^ 0, (pi(x) = 0 for all / implies χ = 0.
In addition to the norm topology, which is also called uniform topology, there
is a host of other topologies in 9Ϊ: The σ(2ϊ, Sl^-topology, called σ-topology for
short, is the weakest topology in 9Ϊ with respect to which all φ e 21* are
continuous. (In the terminology of functional analysis this would be the
w*-topology). On norm-bounded sets the σ-topology coincides with the weak
operator topology. Clearly, a linear functional on 9Ϊ is σ-continuous iff it is
normal.
Let 91^ be the set of all positive normal linear functionals on 9Ϊ. Each φ e 2Ϊ*
defines a seminorm αφ on 9Ϊ by (χφ(χ) = φ(χ*χ)1/2. The s-topology (or strong
topology) in 9Ϊ is the topology generated by these seminorms. In other words, it is
the weakest topology with respect to which all αφ are continuous. It is stronger
than the σ-topology. On norm-bounded sets it coincides with the strong operator
topology. (See Sakai [1971: 20, 33-35] and Dixmier [1957: 32-36].)
The s*-topology in 9Ϊ is the topology generated by the seminorms αρ and a*,
(ρ g 9Ϊ^), where ос* (ζ) = ρ(ζζ*)1/2. Thus, s*-lim l\ = 0 holds iffVlim^ = 0 and
s-lim 4* = 0. On 9lsa the ^-topology and the s-topology coincide.
Recall that the w*-operator topology in J5?(2T) is the weakest topology
which renders the maps Τ -* <φ, Τχ} -·= φ(Τχ) continuous for all φ e 91^ and all
χ e 9Ϊ. An element ρ of 9lsa is called projection if p2 = p. For φ e 91^ the set £φ
= {χ e 9Ϊ: φ (χ* χ) = 0} is a σ-closed left ideal. Since 91 is a von Neumann algebra
there exists a projection qq> with %φ = <Hqq>. qq> is the greatest of all projections q
such that (p(q) = 0. The projection sv = 1 — q(p is called the support of φ. For all
xe 9Ϊ one has φ(χ) = φ(^χ) = φ(χ^). φ is faithful iff ^ = 1. A family Φ
= {φ^ cz 9Ϊ^ is faithful iff %> is the closure of lin{^.§: (pte Φ}. On norm-

270 Special topics
bounded sets, αφ(χη) tends to 0 iff χηδφ tends to 0 in the strong operator topology.
(See Sakai [1971: 31], and Dixmier [1957: 61-62].)
After these preliminaries we can turn to the proof of ergodic theorems.
2. The mean ergodic theorem. A bounded linear operator Те if(9Ϊ) is the
adjoint of a continuous linear operator 7^ e if (91^) (called its pre-adjoint) iff Г is
σ-continuous. We may define T^xp for ψ e 91^ by 7^ψ (χ) = ψ(Τχ), and use the
fact that elements of the bidual Ж** of a Banach space X belong to X iff they are
w*-continuous functional on £*. If Γ is a *-endomorphism of 9Ϊ, then Γ is σ-
continuous because for normal ψ also ψ ° Τ is normal. Thus, *-endomorphisms
have a pre-adjoint.
A linear operator Те if(9Ϊ) is called an αφ-contraction (for φ e 2Ϊ*) if &φ{Τχ)
^ αφ(χ) holds for all χ e 9Ϊ. For Φ с 2Ϊ + we call Γ an ^-contraction when it is an
α^-contraction for all φ e Φ.
Two independent approaches to mean ergodic theorems in 9Ϊ will be described
below. The first works for semigroups of α^-contractions without any algebraic
conditions. The second will be simpler. It relies on Eberlein's theorem and works
for example for d-parameter semigroups of positive operators Γ with Τ^φ ^ φ
for a faithful φ. Only this case will be used in the subsequent treatment of the
almost uniform ergodic theorem.
Theorem 1.1 (Mean ergodic theorem for von Neumann algebras). Let Φ be a
faithful family of normal states for a von Neumann algebra 9Ϊ and let if be a
bounded semigroup of σ-continuous ^-contractions, then the closure of со if in the
w*-operator topology contains an if-absorbing projection P. Px belongs to the
s-closure of со if χ for all χ e 9Ϊ.
Proof (1) First fix some (ре Ф. Clearly £φ = {χ: αφ(χ) = 0} is 5^-invariant.
Therefore, we can define a semigroup if φ = {7^: Те if} of linear operators in
К = «fc, by
Tv(xsv) ··= (Г(х^))^ = (Tx)sv.
We now apply the Gelfand-Naimark-Segal construction: Я^ is a pre-Hilbert-
space with scalar product <x, γ^)φ = φ (у* х) and norm αφ. Let ξ>φ be its
completion. As each Τφ contracts the norm in Я^ it may be uniquely continued to ξ)φ.
Call this extension Τφ. &φ-·= {Τφ: Те if} is a semigroup of contractions in the
Hubert space 9)φ. By the Alaoglu-Birkhoff theorem (2.1.10) there exists an &φ-
absorbing projection Ρφ in if (ξ>φ) and Ρφχ is contained in the α^-closure of
Cx := со §>φχ for any χ e 5)φ.
Now recall that the elements of 9Ϊ are operators in a Hubert space ξ> and that if
is bounded. Put Μ = sup{||r||: Те if}. For хеЯ^ there exists a sequence
zne Cxa {ze Я^: ||z|| ^ Μ\\x\\} with αφ(ζη — Ρφχ) -* 0. For norm-bounded
sequences (xn) in 9Ϊ the convergence to 0 ofxns9 in the strong operator topology is

Von Neumann algebras 271
equivalent to а<Дх„) -* 0, see Dixmier [1957: 62]. Using zn = znsv we find
|| zn ξ — zm ξ || -* 0, (га, η -* oo), for any ξ e £. Put z^ ξ = lim zn ξ. It is then simple
to check that z^ belongs to &φ. Hence, <χφ(ζη — z^) -* 0, and z^ must agree with
Ρφχ. Thus, i^x belongs to the strong operator closure of Cx. Convex sets in У? (§)
have the same closure in the weak as in the strong operator topology, see
Dunford-Schwartz [1958:477].
We have proved that for any хеЯ^ the 5^-fixed point Ρφχ belongs to
σ — со <9^x. It is the unique <9^-fixed point in this set because Я^ is σ-closed and a
second fixed point would be a second fixed point for £?φ in ξ>φ9 contradicting the
uniqueness in the Alaoglu-BirkhoflF theorem.
(2) The second step in the proof is the passage to general Φ. It is unnecessary if Φ
consists of a single φ.
Let us first remark that for projections/?, q the inequality/? rg q means that/? is
a projection on a smaller subspace of ξ> than q9y. (This may be deduced from an
alternative definition of positivity in У? (§): ζ ^ 0 is equivalent to <ζξ, ξ} Ξ> 0 for
all ξ e §, see e.g., Dixmier [1977: 17]). For χ e 2Ϊ and φ e Φ set
Q(p{x)^{ye%:P(p{xs(p)=ys(p and |Ы1^М||х||}.
If φ, ψ have supports s9 й sw then these projections satisfy s^Sy — sv. We obtain
(Tyixs^s^ = (7x)^ = (Tx)s9 = (Τφ(χεφ))3φ. It follows that if у ^belongs
to the convex hull of ^(x^), then ySy must belong to the convex hull of
У<р(х8<р)- As the map и -* w^ is σ-continuous (Dixmier [1957:18]), we deduce
that ysw e σ — со <9^(х^) implies >^ e σ — со 5^(χs^. If ^^ is 5^-fixed, >^ is
5^-fixed. Using the uniqueness proved above we see <2φ(χ) => {2<Дх).
For any φ, tp the inequality ^ rg 5V holds if and only if φ (χ) = 0 holds for all
χ g 2Ϊ+ with tp(x) = 0; (see Dixmier [1957: 62]). As we may assume Φ convex we
see that for any φΐ9 φ2 e Φ there exists ψ (= (φχ + φ2)β) with
Qw(x) c {2^(χ)η β^(χ). The family {(2φ(χ): φ e Φ} is therefore filtered
downwards. Each Qw(x) is non empty by the previous step and w*-compact. It follows
that there exists a point yx in the intersection of all <2φ(χ). As {Tyx-~yx)sv
= Τφ(γχ$φ) — yxsv = 0 holds for all φ, yx is an 5^-fixed point. yx belongs to the <χφ-
closure of со 5^x for all φ. As 9) is the closure of lin {^§: φ e Φ) and as we deal
with norm-bounded sets, yx belongs to the closure of со 5^х in the strong and
hence in the weak operator topology, and thus also in the σ-topology. We have
proved that σ-co 5^x contains an 5^-fixed point for all x.
(3) Finally, let us show that F(^) = {ψ e 2Ϊ*: Τ^ψ = ψ\/Τ# e Sf^ separates
F(Sf) = {u e 91: Tu = иЧТе У}. For any 0 ф и e F(ST) there exists φ e Φ with
νφ i= uSy Φ 0 because Φ is faithful. Clearly νιφ belongs to ξ>φ and is fixed under &φ.
By the Alaoglu-BirkhoflF theorem the condition (iii) in the theorem 2.1.11 is
satisfied for the semigroup §>φ and thus (iv) holds. Hence there exists a
continuous linear form ψ on § with ψ (u^ ф 0 and Τ^ψ = ψ for all Τφ e §^φ. Put ψ (у)

272 Special topics
= \p{ys^). Then ψ is ^-continuous on 9Ϊ and therefore σ-continuous (Sakai
[1971:21]).
It then follows from <7;φ,γ) = <t/>, 7» = <£(7»^> = <y,Z;0^)> =
(T*tp,ys9y = <v>,j^> = ... = <ψ,γ> that φ belongs to F(S?J. Clearly
tp(w) φ 0. As 0 φ и has been arbitrary in F(Sf) the fixed space F{if^) indeed
separates F{if).
Theorem 2.1.11 now implies the first assertion. As σ-co if χ is norm-bounded it
agrees with the closure of со if χ in the weak operator topology. As со if χ is
convex it also agrees with the closure in the strong operator topology. D
Theorem 1.1 is due to Kovacs-Szucs [1966] for *-automorphisms and to
Kummerer-Nagel [1979] for general αφ-contractions with the similar but simpler
proof worked out here.
Corollary 1.2. Under the conditions of theorem 1.1 the closure of coif^ in the
strong operator topology contains an if ^-absorbing projection P#.
Proof. Apply theorem 2.1.11. α
We shall be interested in the particular case of d-parameter semigroups, and use
the notation from § 6.1.
Corollary 1.3. Assume that a commutative d-parameter semigroup
if == {Tu\ueV} satisfies the assumptions in theorem 1.1, then
(α) (Αη)^ψ converges in norm to Ρ^ψ for all ψ e 91^
(b) Px = 5-lim Anx for all χ e 2Ϊ as η -> oo in V.
Proof (a) follows from theorem 2.1.9.
(b) We may assume Px = 0. For any ε > 0 and any φ e Φ there exists a finite
convex combination ΣΜβΜ7^ with οίφ(ΣηβηΤηχ) < β· For и -► oo, \\AnTu — An\\
tends to 0 for all и e V. Hence, for large и,
αφ(Αηχ) й αφ(Σβη(ΑΤη- Αη)χ) -l· αφ(ΑηΣβηΤΜχ)
и
ύΣβη\\ΑηΤ„-Αη\\\\χ\\ + κφ(ΣβηΤ»χ)<2ε. .
Thus, Anxs9 tends to 0 in the strong operator topology of JSf (£>) for all φ. As Φ is
faithful, s-limAnx = 0. α
We now begin with the second approach.
Lemma 1.4. Let Φ be a convex faithful family of normal states in 91^, then the
union U of all order intervals [ — &φ, + Λ;φ] (к e Ν, φ e Φ) is norm-dense in the self-
adjoint part (91 Дл ofW*.

Von Neumann algebras 273
Proof. £/is a linear subspace of (9ϊ^)5α because Φ is convex. If C/is not norm dense
there exists by Hahn-Banach anxe 9lsa with χ Φ 0 and χ(τρ) = 0 for xpeU. For
tp ^ 0 we have
η
χ+(ψ) = sup { Σ *(ψί) + ·· OuWi and Σψί = ψ} ·
i = l
Thus x+(t/0 vanishes for ψ e U. As Φ is faithful, x+ = 0. The same argument
applied to — χ gives x~ = 0. Contradiction, α
Theorem 1.5. Le/ Φ be a faithful family of normal states of a von Neumann algebra
9Ϊ and let У — {Tu: и e V} be a d-parameter semigroup of w*-continuous positive
contractions in 9Ϊ with Τη^φ rg φ for all ueV and φβ Φ. ГАел
(i) (Αη)χψ converges for η -^ со in norm to Ρ^ψ for all ψ e 91^, where P^ is a
positive contraction in 91^;
(ii) a-limn_00 Anx = Px for all χ e 9Ϊ, w/zere Ρ ώ /Ae adjoint of P#;
(Hi) any xe*$lis the s*-limit of a norm-bounded net (χλ) such that each χλ — Px
is a finite convex combination of elements of the form {I — Tu)x.
Proof We may assume Φ convex. A subset D a 91^ is relatively weakly compact
iff it is bounded and lim φ (/^) = 0 uniformly for φ e D holds for every decreasing
sequence /^ of projections with inf/\ = 0, see Takesaki [1979:149]. Thus the
intervals [0, tp] are relatively weakly compact for 0 rg ψ e 91^. If ψ belongs to an
interval [0, k(p~\ with φβ Φ then Αηψ belongs to this interval. By Eberlein's
theorem Αηψ converges in norm for xpeU. (i) then follows from the lemma, (ii) is
a consequence of (i).
(iii) By (ii), Ρ χ belongs to the closed convex hull of У х in the weak operator
topology of У? (§). But this coincides with the closed convex hull in the strong
operator topology and in the s*-topology; Takesaki [1979], p. 70. Hence there is
a net (ζλ) in со if χ with £*-Нт ζλ = Px. If zA is the convex combination Σ аЯм Tu χ,
then χ — ζλ + Ρ χ = ΣαΛΜ(·* — Tux) + Px- χ is the j*-limit of the norm-bounded
net χλ := χ — ζλ + Px. π
3. Almost uniform convergence. We now turn to the non commutative extension
of the pointwise ergodic theorem. The following concept is a suitable
generalization of the notion of convergence а.е.:
Definition 1.6. A sequence or net (xn) of elements of 9Ϊ converges to x0 almost
uniformly if for every normal state ψ and for every ε > 0 there exists a projection
pEtW with v>(a,v) > 1 - β and \\(xn - x0)A.vll -* °-
Remark. If φ is a faithful normal state and pk a sequence of projections with
Ψ(Pk) -* 1> then αφ(1 — p^ = φ(1 — p^) tends to 0. Then s-lim (1 — p^sv = 0 (use
Dixmier [1957: 62] again). By ^ = 1 we obtain s-lim/^ = 1 and hence σ-1πη/\

274 Special topics
= 1. We conclude that the existence of ρεφ for all ε > 0 already implies the
existence of ρεψ for all normal states ψ.
Our aim is
Theorem 1.7 (Almost uniform ergodic theorem). Let ^ = {Tu:ueV} be a d-
parameter commutative semigroup of w*-continuous positive operators in a von
Neumann algebra 9Ϊ with T^l^l, (weV). Assume that there exists a faithful
normal state φ with (Tu)^ φ ^ φ, (и е V). Then, for each χ e 2Ϊ, the net Avx
converges almost uniformly to Px, (v -> oo), where Ρ is the ^-absorbing projection of
theorem 1.5.
The breakthrough work for this theorem was done by Lance [1976], who proved
it for d = 1 and *-automorphisms Tu. Kummerer [1978] obtained the proof for
d — 1 under the present assumptions and Conze and Dang-Ngoc [1978] gave a
proof for d ^ 1 with the added assumption that the Tu are α^-contractions with
Tu 1 = 1 and Τ* φ = φ. Here their arguments will be combined and the case d > 1
simplified.
For the proof of theorem 1.7 we will have to rely on some results on von
Neumann algebras, which go beyond the usual textbook material. They will be
stated as theorem 1.11 and 1.15 below. Moreover, we shall need a basic theorem
from the theory of compact convex sets:
For compact convex AT in a locally convex Hausdorff space let A (K) be the set
of continuous real valued affine functions on K. The barycenter βμ of a
probability Radon measure μ on К is the element of К with f{βμ) = \fdμ for all
fe A (K). For β e К let Ωβ be the set of all probability Radon measures μ with βμ
= β. Forfe C{K) put/= inf {ge A(K): g ^/}. The result we need is:
Theorem 1.8. For anyfe C{K) the function J is concave and upper semicontinuous
on К and satisfies f (β) = sup {\fdp. μ e Ωβ}.
We refer to the book of Alfsen [1971], 1.3.6, for a proof.
If 91 is a von Neumann algebra the state space
S:={tpe9I*:tp^0 and ||ψ|| = 1}
with the w*-topology is compact and convex and we shall apply theorem 1.8 with
К = S. Each χ e 9lsa induces a continuous affine function ξχ: ψ -* ξχ(ψ) ·-= ψ (χ)
on S. Conversely, each w*-continuous affine ξ on S determines a unique χ e 9Ϊ
with ξ = ξχ, because it may be extended to a w*-continuous linear function on
9Ϊ*. To avoid overly heavy notation we identify χ with ξχ.
The first step in the proof of theorem 1.7 is the proof of a maximal lemma. We
shall assume d = 1 for a while. Fix η ^ 1, χ e 9Ϊ, and у > 0, and put Sj = Σ Tu,
u = 0
g = sup{SjX-jyl: 1 ujun}, h = g+ and E= {т/> е S:gO) ^0}.

Von Neumann algebras 275
Theorem 1.9 (Maximal ergodic lemma). Under the assumptions of theorem 1.7
{with d — 1) let μ e Ωφ be a measure with \hdμ = Λ (φ), then \E (x — γ\)άμ ^ 0.
(The existence of such a μ follows from the compactness of Ωφ in the topology
of weak convergence of measures (μ„ -* μ iff \/άμη -* \/άμ V/e С(ЛГ)).
Ргяя/. Put Τ·= Τχ and define an operator Γ on C(S) by
{Τη{φ) = \\Τ*φ\\ΑΤ*φ/\\Τ*φ\\)
for φ with || Γ* φ || Φ 0, and = 0 else. It is simple to check Tfe C(S) for/e C(S).
f is a positive linear operator, and || Τ*ψ \ | = (Τ*ψ)(1) = <Π,τ/>> <; <1,φ> = 1
= || φ || shows that Γ is a continuous operator in C(S). The restriction of Γ to
A (S) coincides with T.
We now argue as in Garsia's proof of Hopfs maximal ergodic theorem: From
T^ 0 and T\ й 1 we obtain
Th=T( sup (Sjx-jyl))^ sup (Щ.х-л1)) +
^ sup (5.+1x-0 + l)yl-(x-yl)) +
= sup (Sjx-jy\ -(x-yl))+ ^g-(x-yl).
Hence
J (x - γί)άμ ^ J fe - ΤΗ)άμ = JArf/i - J ΤΗάμ ^ JArf/i - J ΤΗάμ.
ЕЕ Е
Put α = <Γ1, μ> = <1, Τ*μ}. For α = 0 the assertion of the theorem follows
from \ηάμ — 0. For α Φ 0 the measure ν = α-1 Γ* μ has total mass 1. For any
ZeA(Q),
(1.1) \zdv = α-^Τζάμ = οΓ1^) (<ρ) = ^O*-1 Γ*φ).
The element θ = α-1Τ*φ belongs to S because ||α-1Γ*φ|| = α-1<Γ*φ, 1>
= α-1 <φ, f 1> = α-1 \ΐ\άμ = α-1 <1, f *μ> = 1. As (1.1) holds for all ζ we
obtain ν g Ω^. If Ih denotes the set {у е Α{β)\ у ^ A}, then by theorem 1.8, the
definition of h and Τ* φ ^ φ
\ηάμ — <x$hdv S asup{|Α^/ρ: ρβ Ω$}
= αΑ"(β) = α inf {у(α-1 Τ*φ): у е Ih}
й inf {y(<p): ye Ih} = Η (φ) = \Ηάμ. и
The second step in the proof of theorem 1.7 is a kind oi dominated estimate which
says that for "small" χ e jtf+ there is a "small" common bound for all Anx. This
vague statement is formalized for d = 1 by
Theorem 1.10. Let 9Ϊ, Sf — {Tu},(pbeas in theorem 1.7 with d=\, andO rg ε ^ 1.

276 Special topics
For any xe 2Ϊ+ ш7А||х|| rg 1 αηάφ{χ) ^ ε there exists an element с е 9Ϊ+ w*YA||c||
rg 2 and φ{ό) ^ 4ε1/2 such that Anx ^ с holds for all n.
Before entering the proof recall that a subset F of a convex compact space К is
called ^ face if it is convex and ξ,η e F holds for any pair ξ, η e К for which there
exists 0 < λ < 1 with λξ + (1 — λ)η e F. The face generated by ξ e К is the set
Ρ(ξ)·.= {ηβΚ:3ψβΚ and 0<A<1 with ξ = λη + (1 - λ)ψ}.
Here К will again be the state space S с 2Ϊ*. For a face i7 put
FL = {ze2I**:z^0 and <t/>,z> = 0 VtpeF}.
We shall require the following result, contained in theorem 4.6, 5.2, and 2.4 of
Eflfros [1963]:
Theorem 1.11. IfF с S is a face, its norm closure clF is a face and F1 = (clF)1
consists of the positive elements of a left ideal in 9Ϊ** closed in the w*-topology.
Proof of theorem 1.10. We proceed in three steps (i)-(iii):
(i) First fix n ^ 1 and assume 0 < ε < 1. Put γ = ε1/2, and let g, Α, Ε, μ, φ be as
above, ε < γ and φ (χ) = \χάμ S ε implies J (χ — γ \)άμ < 0. The maximal lemma
then yields μ (Ε) < 1. Set δ = μ (S \ Ε). Let μχ be the restriction of δ _1 μ to S \ £
and let φχ be the barycenter of the probability measure μχ.
Our first aim is to show (A}x — y\) (ψ) S 0 for all ψ e Ρ(φ1) and 1 Sj й п.
Assume this fails. Then we have h(\p) > 0. For ψ e F((px) there exists ξ e S and
0 < λ < 1 with φχ = λψ + (1 — Я)ξ. As A is non negative and concave h{cpx) > 0.
From the way A is defined there exists an/e Α (β) with/^ A and f(cp) < Α (φ)
+ δΗ(φί). Now a contradiction results from
h(cp) = jAJ/i = j" Α^/μ ^ J/rf/i = J/^/i — J /#μ
=/(φ) - <5/(φι) < Α (φ) + δΚ{Ψί) - δ/(Ψι) ί Ε{ψ).
(ii) Now we want to construct a small upper bound cn for AjX, (1 Sj S я): By
theorem 1.11, G-=cl(F{(p{j) is a face and, by continuity, (^лг —yl)(y>) rg 0
holds for φ e G. G1 consists of the positive elements of a w*-closed left ideal in
9Ϊ**.
Such a left ideal is of the form 2Ϊ**# for a projection q; see e.g. Takesaki
[1979: 76]. Clearly, G = {ρ e S: ρ far) = 0}.
Put p = 1 — q. Assume a e 9Ϊ is an element with φ (α) ^ 0 for all xpeG. For any
ξ e S the functional ξΡ(·) = ξ(ρ p) isa, non negative multiple of an element in
G. Hence, ξ (pap) Ξ> 0 holds for all ξ e S. This yields /?ap ^ 0. By applying this
argument to a = yl — ^x we find γρ ^p(Ajx)p.

Von Neumann algebras 277
For any / g 9Ϊ+ we have (p — q) t(p — q) ^ 0. Adding t = (p + q) t(p + q) we
obtain t ^2ptp + 2qtq. Now 0 rg AjX ^ 1 implies
Λ,·χ ^ 2p(Ajx)p + 2q(Ajx)q й 2yp + 2?, (1 ^7^/1),
so that c„:= 2yp + 2# is an upper bound with ||cn\\ S 2.
To estimate φ(0 we use the maximal lemma:
ε-ε1/2 ^ <p(x-yl) = J (х-у1)ф + J (χ - у1)ф
^ <5<Pi (x - у 1) ^ - <5y = - <5ε1/2
implies δ ^ 1 — ε1/2 and therefore φ(ρ) ^ δφ1(ρ) = <5 ^ 1 — ε1/2. Hence φ(0
^27φ(/>) + 2(1-φ(/>))^4ε1/2.
(iii) The sequence cn e 9Ϊ** has a w*-cluster point c^ e 9Ϊ** with || ς^ || :S 2, ^x
^, (l^y<o)), and^O^e1'2.
We now have a bound in 9Ϊ** and it remains to map it into 9Ϊ. Consider the
canonical inclusion к of 91^ into 9Ϊ*. Its adjoint is an order preserving isometry
π = к* of 9Ϊ* * into 91. If ζ is an element in 9Ϊ and ξζ its canonical embedding into
9Ϊ* * (so far identified with z), then π (ξζ) = ζ. We can take с ·-= π (c^) as the desired
bound of all AjX in 9Ϊ.
Finally observe that the assertion of the theorem is trivial for ε = 0 since φ(χ)
= 0 implies χ = 0. Also the case ε = 1 reduces to the case just proved: for
φ{χ) < 1 we may decrease ε a little and for φ{χ) = 1 we have 0 rg χ rg 1 and
φ(1 — χ) = 0 and hence χ = 1. Then take с = 1. α
For general uf^l a dominated estimate is derived by induction:
Theorem 1.12 (Multiparameter dominated estimate). Let 9Ϊ, 9*, φ be as in
theorem 1.7. For xe9l + there exists ce9l+ with \\c\\ ^ 2d||x||,
<p(c) ^ 2d+2||x||1"2"d<p {xf~d and Avx й с for all veV.
Proof. For d = 1 this, and even the better estimate cp(c) rg 4||χ||1/2φ(χ)1/2?
follows from theorem 1.10, (pass to x/||x||). If the theorem has been proved for
some d^\ and Sf'={T^:w'eV'} (with V = {0,1, ...}d+1) is a (d+1)-
parameter semigroup we can define a d-parameter semigroup 9 by Tu = Τ{„0ρ
(и е V), and a 1-parameter semigroup {S°, S1,...} by Sk = ^o,o,...,o,k)? (^ = 0)·
For и/ = (wl9..., wd+1) = (w, wd+1) we have^ = >4Wd+1(S) <>AW. If cis the
element majorizing all Awx and c' the element with Ak(S)c rg c' obtained by
applying the 1-parameter case to c, then Awx <^ c' holds for all w' e V and we have
He'll ^2||c||^2d+1 and

278 Special topics
Ф(0^4||с||1/>(с)1/2
^42άΙ2\\χ\\1Ι2)(2^22\\χ\\^2-2-(ά+1)φ(χ)2-{ά+1))
This is the assertion for d + 1 instead of d. α
For the third and final step of the proof of theorem 1.7 we will need the following
technical lemmas:
Lemma 1.13. If ρ e 2Ϊ is a projection and a,be% satisfy 0 rg a rg b rg 1, /Леи
Proo/. ||α/?||2 й \\αίΙ2\\2\\α1Ι2ρ\\2 й \\рар\\ й \\pbp\\ й \\bp\\. π
Each ζ e 2Ϊ may be uniquely written as ζ = (zx — z2) + i(z3 — z4) with zt e 2Ϊ+
and zxz2 = 0, z3z4 = 0. Put z+ + -=zx + z2 + z3 + z4.
Lemma 1.14. Ifib^) is a bounded sequence in 9Ϊ with £*-Нт l\ = 0, /Леи 5*-lim ^ +
= 0.
Preo/. If bk = hk + ilk with self-adjoint Ak? /k then Ak = {l\ + 4*)/2 shows £*-Ит Ak
= 0. Now \hh\* \hk\ = \hk\2 = htht yields s-lim\hk\ = 0. By A+ = (|Ak| + Ak)/2
we obtain s-lim Ak = 0. Argue similarly for s-lim hk = 0 etc. α
For the next lemma we need the following non commutative Egoroff theorem
due to Saito [1967]:
Theorem 1.15. If(xj) is a norm-bounded net in a von -Neumann algebra 9Ϊ with
я*-Нт χλ = 0, there exists a subsequence (χχ), (j ^1), such that χλ. converges to 0
almost uniformly.
Lemma 1.16. Let 9Ϊ, Sf, φ be as in theorem 1.7. If{xj) is a bounded net in 9Ϊ with
s*-\imxx = 0, there exists a subsequence (xXj) such that Ανχλ. -> 0 almost
uniformly and uniformly in veV as j -> 00.
Proof. First assume χλ ^ 0. Dividing by 2dsup||xA|| we may also assume \\χλ\\
<z2~d. By the Cauchy-Schwartz inequality ελ--= φ(χλ) = φ(1χλ)
rg φ(1) φ(χ*χλ) tends to 0. By theorem 1.12 there exist elements cke 2Ϊ+ with
ψ(€λ) -* 0, \\cx\\ S 1, and Avcx ^ ^„Хд for all ν e V. By 0 rg cA rg 1 we obtain
0 rg с* сх rg с χ and ф(с*сА) -> 0. As φ is faithful and cA self-adjoint, 5*-Нт сл = 0.
Now theorem 1.15 asserts that there exists a subsequence (cx) and for each ε > 0
a projection/^ with φ(ρε) > 1 — ε such that ||ся д || -* 0. Lemma 1.13 then shows
that НС^л^Л II -> 0 holds uniformly for veV.

Von Neumann algebras 279
In the general case we may write χλ = χλί — χλ2 + /(хдз — Хы) as above and
we have 0 rg xXk rg Χχ +? (k = 1, 2, 3, 4), and 5*-lim χ/ + = 0. Applying the first
part of the proof to x~£ + we obtain ρε with ||(^4|7л:^+)д || -* 0 uniformly in V.
Lemma 1.13 now yields \\{Avxkjk)pe\\ й ΙΙ(Λ<+)αΙΓ/2 ^ 0. α
Finally, we can complete the proof of theorem 1.7: We apply theorem 1.5 with Φ
= {φ}. We may assume Px = 0. So there exists a norm-bounded net (xA) with
5*-lim χλ = χ such that each xA is a finite convex combination of elements (/
— Tu)x. By lemma 1.16 there exists for ε > 0 a projection д with φ(ρε) > 1 — ε
such that for δ > 0 there exists some A0 with \\(Av(x — χλο))ρε \\ < δ/2 for all ν e V.
Since xAo is a finite convex combination of elements (/— Tu)x and
\\AV(I — Tu)x\\ tends to 0 for each щ there exists v0eV such that v^v0 implies
\\AV(I—Τη)χ\\<δ/2 for the w's in question. Then H^x^JI < δ/2 and hence
11(Л*ло)ДII < δ/2. \\(Ανχ)ρε\\ < δ follows. We have proved ||(А*)й11 -* 0. α
Remarks. (1) In a similar way it can be proved that for finitely many
x(1),..., xiK) e 21 there exists ρε with φ(ρε) > 1 - ε and ||(Avx(k) - Px{k))pE\\ -* 0
for Λ; = 1,..., AT and t; -* oo.
(2) Again, essentially the same arguments show that || (Avx — Ρχ)++ρε\\ -* 0. By
lemma 1.13 this implies \\(Avs - Ρχ)*ρε\\ -► 0.
Corollary 1.17. Under the assumptions of theorem 1.7 we also have s*A\mAvx
= Px, (v -* oo).
Proof. We may assume Px = 0. Theorem 1.7 implies the existence of a projection
д with <р(д) > 1 - ε and ||(4,*)All -* °- Now
MA*)]2 = φ(\Ανχ\2) = ф([д + (1 -д)] (А*)* (А*) [Д + (1 -д)])
^ (Ка(А*)*(А*)а) + φ((1 -д)|Л*12) + fl>(IA*l2(l -А))
^Ш(А*)а12 II + 21|Л*II (Ф(1 -А))1/2 (by Cauchy-Schwartz).
The first term tends to 0, and φ(1 — ρε) < ε. As ε > 0 was arbitrary αφ(Ανχ) -* 0
follows. Using the second remark above the same proof yields a* (Avx) -* 0. As φ
is faithful this completes the proof, α
Notes
To keep these notes short we refer to Takesaki [1979] for terminology from the theory of
von Neumann algebras. Sinai and Anshelevich [1976] have obtained an almost uniform
ergodic theorem for the case of translations in algebras of local observables, which they
submitted slightly earlier than Lance [1976].
If the semigroup Sf in theorem 1.1 consists of *-homomorphisms, we can identify the
projection Ρ as a conditional expectation operator: There exists a net (Αλ) consisting of

280 Special topics
convex combinations of elements of Sf such that ψ(Αλχ) -> ψ(Ρχ) holds for all ipe^(r
The v4A's are *-homomorphisms and satisfy ΑλΡ = ΡΑλ = P. Hence xp(P((Px)y))
= \imxp(A,((Px)y)) = \imip((Px) (Axy)) = ψ((Ρχ) (Ρу)), (since ψ((Ρχ) ■) e Я J. As φ
was arbitrary, P((Px) y) = (Px) (Py). Thus л; -► /be is the conditional expectation onto
the subalgebra РЧХ of ^-invariant elements.
For this result actually the inequalities T(x*x) ^ (Tx)* 7л;, (ГеУде 2ί), implied by
complete positivity of Sf, are sufficient, see Kummerer-Nagel [1979]. Tischer and
Wittmer [1974] asked which properties of χ carry over to Px, Frigerio and Verri [1982]
considered the case where there is no faithful family of normal states.
Ergodic theorems for continuous parameter semigroups £f = {Tt: t = (tu ..., td), tt ^ 0}
follow easily from the discrete counterparts. Conze and Dang-Ngoc [1978] have proved
the almost uniform ergodic theorem for semigroups {Tt: te G} indexed by elements t of a
connected amenable locally compact group G, i.e., the non commutative extension of the
result of Emerson-Greenleaf. (They assume the existence of a faithful φ, for which the Tt
are positive ^-contractions and satisfy 7^1 = 1 and φ ° Tt = φ.) They also prove local
ergodic theorems in this setting, and even a local ergodic theorem, when G is a Lie group.
FJ.Yeadon [1977] has extended the theorem of Lance in a different direction. He
assumes the existence of a faithful normal semi-finite trace ρ on 2ί and considers a positive
linear map Tin the space Lx (% ρ) of (unbounded) ρ-integrable operators in the sense of
Segal [1952]. Lx may be identified with ЗД*. If 0 ^ Ту ^ 1 and ρ (Ту) й Q(y) holds for
yeLlr\4i with 0 ^ у 5Ξ 1, then for χ e Lx the averages An(T)χ converge to some χ e L1
in the sense that for all ε>0 there exists a projection ρε with ρ(1 — ρε)<ε and
\\ρε(Αη(Τ)χ — χ)ρε\\ -> 0. R. Jajte [1984] has proved a result of this type for subadditive
sequences. He considers a *-automorphism Tin a semifinite von Neumann algebra 2ί with
a faithful semifinite normal trace ρ on 2ί and a sequence (xn) of self-adjoint elements of L1
with xn+k 5Ξ xn + T£xk, (n, k ^ 1), and inf {ρ(χη/η): η ^ 1} > — oo. He shows that xjn
converges in Lx and almost uniformly (in the sense of Yeadon) to a 7^-invariant xe Lv
Let φ be a faithful normal state in a von Neumann algebra 2ί and X the closure of 2iSfl in
L2(2i, ψ). Let Γ be a positive linear mapping of 2ί into itself with T\ <Ξ 1 and φ ° Τ 5Ξ φ.
Then Γ can be extended to the complexification X of the closure X of 2iSfl in L2 (2ί, φ). For
iel, Gol'dshtejn [1981] showed the L2-convergence and the almost uniform
convergence of An(T)x. He also showed convergence of Αη(Τ*)ψ to some ψ0 in the norm of 2ίψ
for ipe^, and that, for ε > 0, there is a projection p with φ(/?) ^ 1 — ε and
\ιπιη^ηρ{\Αη(Τ^ψ(γ)-ψ0(γ)\φ(γ)-1: yepWp,y > 0}] = 0.
He uses the following maximal estimate: For positive ε,/s and for (xn) с 2ί+ with
βι=Σεη1φ(χη)< 1/2 there exists a projection /?e2i with φ(ρ)7>1—2β and
||/?(^Μ(Γ)χπ)/?||<2επ,(«,^£Ν).
Gol'dshtejn's proof of the almost uniform ergodic theorem avoids some of the tools
needed here. Jajte [1983] has generalized and simplified the argument.
Luczak [1984] considered a C*-algebra 2ί0 which is ultraweakly dense in a von
Neumann algebra 2ί and a *-automorphism Τ of 2ί. He studied the convergence of
N-l
Ν'1 Σ Τηψ for "almost all" states ψ of 2ί0, where Tn are induced by Tn; see also Radin
i = 0
[1978].
Yeadon [1980] also studied mean ergodic theorems in non commutative Lp-spaces.
Groh and Kummerer [1982] have proved mean ergodic theorems for semigroups of

Entropy and information 281
"bicontractions" in 2ί. These may be regarded as non commutative generalizations ofLi
— L^ -contractions.
Corollary 1.17 was observed jointly with M. Lin.
Ayupov [1977] announced ergodic theorems in 0*-algebras.
Further references related to this section are Abdalla-Szucs [1974], Archbold [1976],
Doplicher-Kastler-Stormer [1969], Saito [1974], Lau [1976], Dang-Ngoc [1982], Radin
[1973], [1976], Watanabe [1979], Szucs [1983], Ayupov [1982].
§ 9.2 Entropy and Information
We now present the theorems of Shannon-McMillan and Breiman-Ionescu-
Tulcea and their multiparameter generalizations.
1. Preliminaries and mean convergence. (Ω, stf\ μ) is a probability space and 0>
the family of partitions ξ = {Ah i e 1} of Ω into finitely many or countably many
disjoint measurable sets At. We write ξ = {At} for short. If η = {Bj} is a second
partition ξ ν η denotes the common refinement, i.e., the partition consisting of
the sets АьглВ}. η is called finer than ξ if ξ ν η = η. We then write ξ rg η. ξ is
identitified with the σ-algebra generated by ξ. In particular μ (Α \ ξ) = Ε(ίΑ\ ξ) is
the conditional probability of A given ξ.
The conditional information of ξ given a sub-a-algebra ^ cz si (or given a
partition) is defined by
/«l*) = - Σ \ΑΜμ{Α^).
(The choice of the logarithm is arbitrary. Let us take the natural logarithm. We
put log 0 = - oo, 0 log 0 = 0.)
The conditional entropy Η {ξ \0$)οϊξ given M is the expectation of I {ξ \ Ж). The
information 1{ξ) of ξ and the entropy Η (ξ) are the quantities Ι(ξ\&), Η(ξ\&)
obtained when Μ is the trivial σ-algebra {0,Ω}. Clearly,
Η(ξ) = -Σμ(Αί)Ιοζμ(Αί).
Proposition 2.1. For ξ,η,ζβ 0> we have
(ϊ)Ι(ξνη\ζ) = Ι(ξ\ζ) + Ι(η\ζνζ);
(ii) Η(ξνη\ζ) = Η(ξ\ζ) + Η(η\ξνζ);
(Hi) ξ й η implies Ι (ξ | ζ) й I (η 10 and Η {ξ \ ζ) й Η {η \ ζ).
Proof (i) is obtained by a direct computation, (ii) follows taking expectations,
and (iii), since all terms are non negative, α
Lemma 2.2 (Chung-Neveu). If^ cz (€2 cz ... is an increasing sequence ofsub-σ-
algebras of si and ξ e 0>, then
Ε(*νιρΙ(ξ\%))^Η(ξ) + ί.

282 Special topics
Proof. Put /* = supn/(^|^), and If = sup^-l^log/^K)). For t > 0
consider μ(Ι? > t) = μ(ω e A{. ΪΏΪημ(Αί\%^ (ω) < e *).
The sets Ain = {ω: μ(Αί\%) (ω) Ξ> e_i for к < n and μ(Αί\%) (ω) < е-'} are
^„-measurable. Summing the inequalities ^1Α.ημ(Αί\(^η)άμ ^ e~*μ(Αίη) over л
we obtain μ(Ι* > t) й ^~ίΣ,ημ{Αίη) ^ <?"'. Hence μ(Ι? > t) <Z Min(e_i, μ(Αί)).
As {/* > /} is the disjoint union of the sets {If > t) the desired estimate follows
from
00
£(/*) = Ι μ(Ι*>ήώ
о
00 00 00 00
= Σ ί μ(!? > i)dt й Σ J Min(e-f, μ(Α$άί
i=l 0 i = l 0
оо
= Σ 1-μ(ΑΪ 1ο§μ(Λ;) + μ(Α^ = Η (ξ) + 1. □
ΐ=1
Lemma 2.3. Le/ D be a denumerable directed set and {3$u, ue D} an increasing
family of sub-o-algebras of si. If ξ is a partition with Η (ξ) < oo and3S = \] 3tu the
σ-algebra generated by all 3$u, then Ι(ξ\&„) converges to Ι(ξ\3$) in Lx-norm.
Proof Otherwise there exists an increasing sequence ul<u2<u2< ... inD and
an ε > 0 with \\Ι(ξ\&Η1) — Ι(ξ\&)\\1> ε, such that for all и в D there is some
uk ^ и in the sequence. The sequence (€k ··= 3$Uk is increasing and generates 38.
Therefore the conditional probabilities μ (At | ^k) tend in Lx and stochastically to
μ(Α{\38). Hence logμ{Α{\^k) tends stochastically to logμ(Αι\3$) and Lemma 2.2.
implies ||/(^|^k) - /(£|#)Hi -* 0, a contradiction, α
We now consider multiparameter groups and semigroups of measure preserving
transformations in Ω. We begin with the invertible case. Let d ^ 1 be a fixed
integer and {τΜ: // e Zd} a group of automorphisms of (Ω, .я/, μ). For
ξ = {AUA2,... j, i(w) denotes the partition τ"1 ξ = {τ~χ Αχ, τ~ι Α2,...}, and,
for Wc=:Zd, ξ(Η) denotes the partition or σ-algebra VweiriW· For
w = (w1?..., wd), t; = (/"!,..., vd) we again write и ^v, when wf ^ t;i? (/ = 1,..., d),
and w < ν when wf < vb (i = 1,..., d). [w, v\_ is the "interval" {w: w <; w < f}.
w < e ν shall mean that и is smaller than ν in the lexicographic order. Put
0 = (0, 0,..., 0) and e = (1,1,..., 1). 7r(w) = wx · u2 ... wd is the number of
elements of [0, u\_. и -* oo shall mean that all coordinates of w tend to oo. The
following theorem has its roots in the fundamental work of С Shannon [1948],
which contains the proof of stochastic convergence for d = 1 and finite ξ.
McMillan [1953] proved Lx-convergence in this case and Carleson [1958] for d
= 1, Η (ξ) < oo. The multiparameter extension was given by Fritz [1970], and
rediscovered by Thouvenot [1972], Katznelson and Weiss [1972], and Follmer
[1973].

Entropy and information 283
Theorem 2.4. (Mean ergodic theorem for information). Let {Tu:ueZd} be a
group of automorphisms of a probability space (Ω, «я/, μ) and Η (ξ) < oo. Then
n(w)~11(ξ([0, w[)) converges in L^norm for w->oo. The limit is
h--= Ε(Ι(ξ\ξ({\ν: w <<fO}))|e/) where J is the σ-algebra of invariant sets. In the
ergodic case the limit is Η{ζ\ξ{{\ν\ w <^0})).
Proof For ue [0, w[ putFuw = {ve [0, w[: ν <eu\. Repeated application of
lemma 2.1. (i) gives
(2.1) /(f([0,W[))= Σ /({(n)|f(/£,w)).
M6[0,w[
As the transformations τΜ are measure preserving and invertible we have
μ{τ~χΑ\Β) = μ(Α\τ„Β). It follows that
(2.2) muM{FUiW)) = im(FUiW-u))oxu
where F— и = {ν — и: ν e F}. The set D of all pairs (w, w) with w > 0 and
w e [0, w [ is a directed set with the partial order ^ d given by (w, w) :g d (r, 5) iff
Fuw — uc=:FrtS — r, and the family of σ-algebras &{UfW) -·= £{FUiW — u) is increasing
and generates Μ = ξ ({ν: ν <^0}). ((w, w) is large in the partial order ^d if
Μ (и, w) — Min {w1?..., ud, wx — uu ..., wd — ud} is large. If M(u, w) exceeds η
then Fuw — u contains all ν < € 0 with | v{ | rg η — 1, (ι = 1,..., d)).
Lemma 2.3 implies that for any ε > 0 there exists an η with
||/({|{(/£.w - и)) - /(fim < β for (и, W) g C?n,w:= {(и, w): M(u, w) ^ n).
For fixed n, n(w)~x card (С?пм;) -> 1, (w -> 00). This, and the boundedness of the
integrals ||/(£|£CFMiM; - n))||j й Η(ξ) + 1 (lemma 2.2), implies that
IItt(w)-1 ς i^\m^u-n{wy' Σ /(^.w-«0)°*Jli<2e
«e[0,w[ ме[0,м>[
holds for large w. The desired result now follows from (2.1) and (2.2) since
π(νν)_1ΣΜ6[ο,νν[^(^Ι^)οτΜ tends to h in L1-norm by the mean ergodic
theorem, d
It is an exercise to show that the limit is Ε(Ι(ξ\ξ(\Λ9αο[))\^) for d = 1, and
Η(ξ\ξ([ί9αο[)) in the ergodic 1-parameter case.
The поп invertible case. We sketch the proof that theorem 2.4 implies the
analogous result for semigroups of non invertible transformations by a passage to the
multiparameter analog of the "bilateral extension". The index set now is
V = {0,1, 2, ...}d. {tu: ueV} is a semigroup of endomorphisms of Ω with
τ0 = identity.
Let ξ = {Al9 A2,...} with Η(ξ) < oo be given. Put Xu(co) = i if ω e τ"1 Αχ.
Let μ! be the measure on {1,2, ...}v = Ω' which is the distribution of

284 Special topics
X = (Xu: и e V). In other words: if V0 is a finite subset of V and (ω'),, the f-th
coordinate of α/ β Ω\ then
μ'({ω': (ω')ν = iv for ν e V0}) = μ({ω β Ω: Χν(ω) = iv for v e V0}).
μ' may be extended in a unique way to a measure on Ω" = {1, 2,...}Zd which is
invariant unter the shift transformations 0M, (и е Zd). If ξ" is the partition of Ω"
into the sets {ω": (ω")0 = ι}, (ι = 1, 2,...), then the joint distribution of the
family {/(^([0, u[)): ue V} on Ω (under μ) is the same as that of the family
{/(<T([0, u[)): и e V} on Ω" (under μ"). Applying theorem 2.4 to ξ" we obtain
the Lx-convergence of n(u)~xΙ(ξ([0, u[)) for w -* oo.
2. Convergence a.e. The following theorem was proved for d — 1 by L. Breiman
[1957/60], and A. Ionescu Tulcea [I960], and for general d(assuming ergodic-
ity) by Ornstein and Weiss [1983].
Theorem 2.5 (Pointwise ergodic theorem for information). Let {τΜ: и е V} be а
d-parameter semigroup of endomorphisms of Ω. If ξ is a finite partition
η~άΙ(ξ([0, ne[)) converges a.e. for n -> oo.
Proof. Let ξη be the partition £([0, ne[). If η is a partition, then η (ω) shall be the
element of η which contains ω. By definition Ι(ξη) (ω) = — log μ(ξη(ω)). Let q be
the number of elements of ξ = {Al9..., Aq}.
Lemma 2.6. A* := sup„^x {η~άΙ(ξη)} is integrable.
Proof. For t > 0 put Дв:= {ω: - л~<* log μ (ξ" (ω)) > ί}. Д" is the union of those
elements 5 of the partition ξη which satisfy μ{Β) < e~tnd. As ξ" has at most qnd
elements, we have μ(Β?) < {qe~yd. Hence μ (A* > i) ^ Ев^О"* й e~t/2 for ί
00
large, say, for t ^ t0. The integrability follows from J Α* άμ = J μ (A* > t)dt rg /0
о
00
+ J e~t/2dt<oo. α
ίο
Now set Α(ω) = lim sup H"d/(<i;n) (ω) and Α(ω) = liminf H~d/(£n) (ω). By
lemma 2.6 both A and h are finite a.e.. Let el be the element of V with i-th
coordinate 1 and all other coordinates 0. As ξη+1 is finer than τ'ί1 ξ" we have
Ι(ξη+ί) ^ /fc1 £w) = /(ξ") °_τβ<. As (л + l)d//id tends to 1 this implies h ^ A"o v
and A ^ Α ο τβζ. Since A and /Tare integrable we obtain A = Α ο τ£ί and A = Α ° τ£ί.
Thus A and /Tare invariant under all τΜ, {и е V). If almost sure convergence fails,
there exists oc> β and an invariant set F of positive measure with A > β and A < α
on i7. This means that for ω e F there are infinitely many n with μ(ξη(ω))
> exp( — nda) and infinitely many n with
(2.3) μ(ξ»(ω))<εχρ(-ηάβ).

Entropy and information 285
Put Q = {1, ...,#}. For Wcz V the elements of Qw are called f^-names and the
f^-name of ω is the element у = (iu)ueW of Qw with τΜω e ^iu for all ме Ж. By
abuse of notation μ (у) := μ {ω: the f^-name of ω is у} is called the measure of the
name y. Wis called the domain of y. If the domain is a "cube" [w, w + ле[ for
some w e V and л ^ 1, we call у an «-name and η its side-length. We call у fat
when μ(γ) > exp( — ccnd). ξη(ω) is called fat, when the [0, ne [-name of ω is fat,
i.e., when μ(ξη(ω)) >exp( —and).
Say that a name у is ε-covered by names {y,·} if the union of the domains Wj of
the y,- contains all but a fraction < ε of the points of the domain W of y, and if
each y, agrees with у on Wj η W. If, moreover, the Wi are contained in W and
disjoint we say that {y,·} ε-ΐί/α? у.
Lemma 2.7. For all E>Q,m^l and for almost all сое F there exists TV (ω) such
that for TV ^ TV(co) the [0, Ne[-name of ω can be ε-tiled by fat names of side-length
^ m.
Proof Given any ε', ε" > 0 and К ^ 1 we can inductively find numbers m
= mx < nx < m2 < n2 < ... < nK as follows: Put mx = 1. When mfc has been
determined find nk > mk such that the set Fk of points ω e F having a fat [0, ve[-name
for some ν with mk :§ ν rg nk has measure ^ //(i7) — ε'/ΛΓ. Then find mk+1 with
nk/mk+l <ε". The intersection i7' of the sets Fk, (k = 1,..., AT), has measure
^ //(i7) — ε'. If ε' is very small most of the points of the invariant set F visit F'
most of the time. Formally: Given εχ > 0 we assume ε' < ε\β. The limit/of the
sequence/^ = N~dYdUe{0Ne{ \F, ° τΜ exists a.e., (theorem 6.1.2 or 6.2.8) and
satisfies Ο^Τβίρ and J/rf/i = /i(F). Now μ({ω_Ε Fj((o) й 1 - fii/2}) · (eJ2)
^JOf- ί)άμ<ε' implies that F" ··= {a>e F:J(<d)>\ — ε^β) has measure
For ω g i7" there exists TV (ω) with TV ^ TV (ω) =>/Ν(ω) > 1 — εχ. We may
assume that TV (ω) is much larger than nK. As F" was an arbitrarily large subset of F
it will be enough to show TV (ω) has the desired property for small enough ε1? ε"
and large K. We can then repeat the argument with F replaced by F\F" and
smaller εχ. First choose AT with (1 — 3~{d+1))K< ε. We may assume εχ < ε/4. Let
RK+i denote the set of и e [0, Ne\_ with τΜω e i7'. i?x+1 covers 1 — εχ of [0, TVe[.
Put Sk+1 = [0, Ne[\RK+1. Let S^ be the set of w's in [0, TVe[ for which
\u,u + nKe\_ is not contained in [0, Ne\_. Choosing TV (ω) large we may assume
card№)/TVd < ε/8. Put SK = SK+i и S'K and RK = [0, Ne[_\SK. Then
5K := TV~d card (SK) ^ ε/4 + ε/8 й β/2 - ε/8. For any и в RK and к = 1,..., Л:
there exists a cube [w, и — ve\_ with side-length between mk and ик such that the
[w, w + ve[-name of ω is fat. (£k denotes the set of these cubes. We shall now use
the Vitali covering argument to pick a disjoint subfamily of these cubes, which
covers most of [0, TVe[. For С = [и, и + ve[putC = [u — ve, и + 2ve[. wis called
the lower corner of С Let CK1 be one of the cubes in &K with maximal side-
length, then CK2 a largest cube in (iK among those for which the lower corner is in

286 Special topics
RK\CKl9 then CK3 a largest cube in (iK with lower corner in RK\(CK1 и СК2), etc.
When no such cube is available any more the chosen cubes are disjoint, and - as
we shall see in a moment - they cover 3"(d+1) of [0, Ne\_.
Now assume 1 rg к < K9 and assume that the choice of disjoint cubes in
&K9..., (£k+1 has been completed. At this point [0, Ne[ is the disjoint union of a
set Rk+1 с RK9 a set Sk+i з SK with sk+1'.= N~d card(Sfc+1) й s/2-s2k~K~29
(fork rg K— 2), and of disjoint cubes selected from (£k+2, (£k + 3,..., (£χ. Let Sk be
the set of w e [0,7Ve[ for which [u9 u + nke\_ intersects one of the disjoint cubes
chosen from dk+l9..., dK. As their side length is 2> rak+1 ^ (β")"1 % we see that
N~d card (S'K) й s2~{K + 3) holds provided ε" is small enough. (The choice of ε"
depends only on К and ε). Put Sk = Sk+1 и 5к and let i?k be the set of points in
[0, Ne[\Sk, which are not yet covered by one of the disjoint cubes. Then
i?kc: i?k+1, and sk>=N-d card(Sk) й | -β*"*"3.
Now we choose disjoint cubes Ck_1?... from dk with lower corners in Rk
as above. Let Uk be the union of the disjoint cubes from ЪК9...9Ък9
uk = 7V"dcard (£/k), and rk = 7V-dcard (Rk). In the case uk+1 ^ 1 — ε we had
covered 1 — ε of [0, Ne[ even after the previous step. In the case uk+1 < 1 — ε the
fact that [0, Ne\_ is the disjoint union of Uk+1, 5к and i?k implies rk ^ ε/2. Hence Rk
contains at least half of the points left uncovered by Uk+l. On the other hand Rk is
covered by Ckl и Ск2 и.... Hence 3d(uk - uk+1) ^ rk ^ (rk + t;k)/2
^ (1 - nk+1)/3. This implies (1 - 3"<d+1>) (1 - Mjk+1) ^ (1 - uk).
Therefore at each stage the uncovered portion is reduced by a factor
(1 — 3"<d+1)) until 1 — ε of [0, Ne\_ is covered. The assertion now follows from
(l-3-(d+1))*<e. α
Now fix m ^ 1 and let ΓΝ be the set of TV-names which can be ε-tiled by fat names
of side-length ^ m.
Lemma 2.8. Ifm is sufficiently large and ε > 0 sufficiently small, then card (ΓΝ)
S exp(7Vd(a + β)/2) holds for all sufficiently large N.
Proof. To specify a name in ΓΝ one can first choose disjoint cubes W} of side length
^ m covering (1 — ε) of [0, Ne\_9 then specify a fat name for each domain Wj and
finally fill in the name arbitrarily off the chosen cubes. If there are /cubes Wj9 then
i ^ Ndm~d because of the disjointness. Each Wj = [u(j)9 v(j)\_ is determined by
its lower corner u(j) and upper corner v(j). There are ( . I possible choices of
/Nd\
the set {u{\)9..., u(i)} and I . I choices of {f(l),..., v(i)}. Hence the total
' fNd\2 ( Nd \
number of ways of specifying exactly /cubes W: is rg ^ , ч with
3 \i ) \M(N9 m)J

Entropy and information 287
M(N, m) — [Ndm d] + 1. The number of ways of choosing the cubes thus is
/ Nd \
S Ndm d ( r,^r \. Take the logarithm, divide by Nd, use Stirling's formula
\M(N, m)J
and let N-+00. Then the resulting expression tends to —m~d\ogm~d
— (\ — m~d) log (1 — m~d), which is smaller than (β — α)/8 for large m. For such
an m the number of choices of the Wj is rg exp (Νά(β — α)/8) for large N.
As the Wj cover (1 — ε) of [0, Ne\_ there are only qeNd — exp (Nds log q) ways to
fill in the names in the uncovered places. Taking ε < (/? — α)/8 \ogq this is
^exp(7Vd(j8-a)/8).
The number of fat names one can put into a cube W} of side-length nd is
^ exp (and) = exp (a card (Wj)). Hence the number of fat names one can put into
a fixed choice Wi9 W2 ... of the disjoint cubes is rg exp(7Vda). Putting all this
together the lemma is proved. D
The remainder of the proof of theorem 5.1 is easy. Let BN be the set of names in ΓΝ
which have measure < exp ( — Nd(/? — (β — α)/4)). Then the measure of the set of
points ω with [0,7Ve[-name in BN is at most card (ΓΝ) exp (— Nd (β — (β — α)/4))
й exp (- Νά(β - α)/4). By Borel-Cantelli the [0,7Ve[-name of almost all ω
belongs only finitely often to BN. On the other hand lemma 2.7 implies that the
[0,7Ve[-name of almost all ω e F belongs to ΓΝ for sufficiently large TV, and hence
to ΓΝ\ΒΝ. This means that eventually, for almost all coeF, μ(ξΝ(ω))
^ exp (- Νά(β -(β- α)/4)), contradicting (2.3). D
Remark. The same argument works when [0, Ne[ is replaced by [0, w[ and
w -> 00 in a sector of V. For groups {τΜ, ue Zd} it also works with squares
[ — Ne, Ne\_, etc. However, it is not clear if the condition that w -* 00 in a sector of
V can be dropped. The generalisation to countable partitions ξ with Η(ξ) < oo
seems to be an open problem for d^2, but the case d = 1 has been proved for
partitions with Η (ξ) < oo by K.L. Chung [1961].
Notes
The original motivation of the Shannon-McMillan theorem has been the study of the
content of information provided by the observation of a random sequence of "letters"
from a finite alphabet. The book of Pinsker [1960] gives an account of the work done by
Shannon, McMillan, Pinsker, Dobrushin, Perez, Chintchin and others on "information
stability" and the "speed of generation of information". (This is apart from the main body
of information theory, dealing primarily with channels, see e.g. Gsiszar-Korner [1981]).
Perez [1957] deals with more general alphabets by considering two probability
measures P, Q on a measurable space (Ω, л/), and a sequence Xl,X2,... which is stationary
both under Ρ and Q. Assume that Pn, Qn are the restrictions to the σ-algebras stfn generated
by Xu ...,Xn and Pn <^ Qn for all n, with dPJdQn =/„. If limn'1 \\ogfndPexists and is
finite and the Xl9X2,... are independent under Q, then limn'1 log/„ exists in the Li-
sense. The theorem of Shannon-McMillan is the special case where the Xi have the
uniform distribution on a finite set under Q. A. Ionescu Tulcea [1960] actually proved а.е.-

288 Special topics
convergence in this generalized setting. S.C. Moy [1961] proved Li-convergence and a.e.-
convergence when XUX2,... is Markovian with stationary transition probabilities; see
also Perez [1964] and Kieffer [1973].
Let us now return to the formulation with partitions. Jacobs [1959] has proved that the
theorem of Shannon-McMillan remains true if the condition of invariance of the measure
μ is replaced by the condition that an invariant probability measure ν with μ <ζ ν exists.
This can be achieved under assumptions of almost periodicity. Parry [1963] has an
extension of the Breiman-Ionescu Tulcea result to conservative nonsingular transformations
having no invariant measure. Then и-11(ζ([О, «[)) must be replaced by suitable ratios as
in the Hurewicz case of the Chacon-Ornstein theorem.
Follmer [1973] had already proved the multiparameter pointwise ergodic theorem for
information in the special case where ζ is the partition generated by the O-coordinate of a
Gibbs field with finite state space. Kunsch [1981] has extended this to countable state
spaces.
Kieffer [1975] has a generalization of the Shannon-McMillan theorem for the action of
amenable groups. Moulin-Olagnier [1983] has a simpler proof of this result based on his
"almost subadditive" mean ergodic theorem for processes {FA} indexed by the finite
subsets A of an infinite amenable group G for which there exists о 0 such that, for each A
m m
and each decomposition 1A = Σ af 1A. with af ^ 0 one has J (FA — Σ af FA.) + ф^с^ at.
Derriennic [1983 a] has a different almost subadditive ergodic theorem which yields a
proof of the Breiman-Ionescu-Tulcea theorem; see § 1.5.
PitskeP and Stepin [1971] proved a pointwise ergodic theorem for information for the
group of dyadic rationals mod 1. PitskeP [1980] has studied the asymptotic equipartition
property for space homogeneous processes. (In contrast to what he writes, his results do
not contradict KrengePs entropy computation since his TA is not the induced
transformation). For further related work and references see PitskeP [1971], [1978] and Nguyen-
Zessin [1976], [1979], and TempePman [1984].
§ 9.3 Nonlinear nonexpansive mappings
Let 9) be a real Hubert space and С a closed convex subset of §. A mapping
T: С -+ С is called nonexpansive if || Tx - Ту \\ й II x - y\\ holds for all x,yeC.T
need no longer be linear. The nonlinear ergodic theorem of Baillon [1975] asserts
that n~x(T° + T1 + ... + Tn~x)x converges weakly if Τ has a fixed point. A
sufficient condition is the'boundedness of С Strong convergence can be obtained
under supplementary conditions. We discuss this material in the multiparameter
setting. Generalizations dealing with more general Banach spaces will only be
reported.
1. Weak convergence of Cesaro averages.
Lemma 3.1. Let С a ξ) be closed and convex and let T be nonexpansive. If
w-limxw = ρ and xn — Txn -* 0 (strongly), then ρ = Тр.
Proof. For any zeCwe have ||z — xn|| ^ \\Tz— Txn\\9 and hence ||z — xn||2

Nonlinear operators 289
^ (Tz — Txn, ζ — x„>. This implies 0 rg <z — Tz — (xn — Txn), ζ — xn}. The
assumptions on xn now yield <z — Tz, ζ — ρ} ^ 0. For 0 < λ < 1 put zA = (1 — λ)ρ
+ Я Γρ. Substituting ζλ for ζ we find <ζΛ — 7ζΛ, /? — Τρ} ^ 0. Now let λ tend to
0. α
If ζ e С is a convex combination of two fixed points x, у of Г and Tz Φ ζ, then Tz
cannot lie on the line connecting χ and у because this would imply
\\Tz-Tx\\>\\x-z\\ or ||7z-7>||>||z->>||. Hence ||x -z|| + \\z - y\\
= \\x-y\\<\\x-Tz\\ + \\Tz-y\\ = \\Tx-Tz\\ + \\Tz-Ty\\£\\x-z\\
+ || ζ — j; ||, a contradiction. Thus, the set of fixed points of a nonexpansive Tis a
closed convex (possibly empty) subset of С
After these remarks we can enter the discussion of semigroups of nonexpansive
mappings. Let d ^ 1 be a fixed integer, V = {0,1, 2, ...}d, and let ¥ = {Tv:veV}
be a semigroup of nonexpansive maps Tv: С -+ C. We shall use the notation from
§6.1 and write 0 = (0, ...,0), Su = Σ^[ο,Μ[^> π{μ) = ux-u2 ...ud, Au
— π{ύ)~ι Su. T0 shall be the identity /. The equation Tu + V= TUTV describes the
semigroup property, и -> oo means that all coordinates of и tend to oo.
If there exists one element xxe С for which the orbit ^xx is norm-bounded,
then ||^x2ll — ll^*iII = II Tux2 — Тих^\\ <; ||x2 — xx\\ shows that the orbit of
any bounded subset of С is bounded. Then the norms of the averages Aux and
also the norms of vectors like TwAux are bounded: there exists Mx< oo with
\\Aux\\, H^xll, ||7;Λ·χ|Ι UMX for all wand we V.
Let e{i) denote the i-th unit vector in V.
Lemma 3.2. If the orbit of an element of С is norm-bounded we have
\\Aux - Te{i)Aux\\ -+ 0, (u -+ oo).
Proof For any ye С and ueVwe have
0u\\Tvx-y\\2-\\Tv+e{i)x-Teii)y\\2
= \\Tvx-Te(i)y\\2-\\Tv+eii)x-Te{i)y\\2
+ 2(Tvx - Te(i)y, Te(i)y-y} + || Te(i)y - y\\2.
Take the sum over all ν e [0, u\_ and multiply by π (и)"1. Most of the first and
second terms cancel each other. We can omit the negative terms not needed for
the cancellation. Writing Ц(и) — {ν = (vl9 ...,vd)e [0, и[: vt = 0}, we arrive at
Ο^τφΓ1 Σ II Tvx-Te(i)y\\2 + 2<Aux-Teii)y,Te{i)y-y}+ \\Te(i)y -y\\2.
veVdu)
Put у = Aux, then <...,...> = — || Te{i)y —y\\2 and we obtain
\\Aux-Te{i)Aux\\2 й n{u)~l Σ \\Tvx-Te(i)Aux\\2.
veVi(u)

290 Special topics
As || Tvχ — Te{i)Aux\\2 is bounded by 4Mj and V{(u) has %(и)\щ elements the right
side is bounded by 4М2^-1. α
The set F= {x e C: Tux = χ for all и е V} of fixed points of <£f is closed and
convex.
Lemma 3.3. If the orbit of an element of С is norm-bounded F is поп empty. If
u{\) < и(2) < ... is an increasing sequence V in V andAu{k)x converges weakly to
p, then pe F.
Proof As {Aux: и е V} is norm-bounded, it is enough to prove the second
assertion. Lemma 3.2 shows that xn-.= Au{n)x satisfies w-limxn = ρ and
\\xn— Te{i)xn\\ -* 0. By Lemma 3.1, ρ — Te(i)p. As i was arbitrary and the Te(i)
generate Sf we have pe F. α
If F is non empty Ρ shall denote the projection on F. This means that Px is the
point of F closest to x. For ζ e C\F the set F is contained in the half space
{yeb:(y-Pz,z-Pz}uO}.
Lemma 3.4. If Ρή= 0, then PTux converges strongly for и -> oo.
Proof For и rg ν we have
\\PTvx-Tvx\\ й \\PTux-Tvx\\ = \\Tv_uPTux-Tv-uTux\\ ^ \\PTux- Tux\\.
The function ν -+ \\PTvx — Tvx\\ is therefore nonincreasing on V. For ε>0
we can find w(s)eV such that w(s) ^ и ^ ν implies 0 rg \\PTux— Tux\\2
-\\PTvx-Tvx\\2<8.
Note that for any у e Fandze С we have \\z — y\\2 = ||z — Pz\\2 + \\Pz — y\\2
+ 2(Pz-y,z-Pzy ^ ||ζ-Ρζ||2 + ||Ρζ->>||2. If we put у = PTux and ζ
= Γυχ we obtain
||ΡΓυχ-ΡΓΜχ||2 = ||Ρζ-^||2^||ζ-^||2-||ζ-Ρζ||2
= \\Tvx-PTux\\2-\\PTvx-Tvx\\2<8.
Hence, ΡΤνχ is a (generalized) Cauchy sequence and converges, α
We can now prove the multiparameter form of the non linear ergodic theorem of
Baillon. It is a special case of a result of Brezis and Browder [1977].
Theorem 3.5. If Sf is a semigroup of nonexpansive maps of a closed convex subset
С of a Hilbert space 9) and F φ 0 (or - equivalently - the orbit of at least one
element of С is norm-bounded), then Aux converges weakly to some ρ e F,(u -> oo).

Nonlinear operators 291
Proof. Let q be the element of F with || PTux — q || -> 0. It will be enough to show
that if u{\) < u{2) < ... is a subsequence V of V for which Au{n)x converges
weakly to some p, then ρ — q.
Summing the inequalities
(p -q,Tvx- PTvx} й <PTvx -q,Tvx- PTvx}
u\\PTvx-q\\\\Tvx-TPvx\\u\\PTvx-q\\\\x-Px\\
over all ν e [0, w[, and dividing by n(u) we obtain
(p-q^AvX-niuy1 Σ ΡΤ,χϊ^πΜ-^χ-ΡχΙΙΣ \\PTvx-q\\.
v<u v<u
The right hand side tends to 0 for и -* oo. Passing to the limit in V and
observing \\ii{u)~l^v<uPTvx — q\\ -* 0 we find (p — q,p — q} й 0. Hence, ρ
= q. Π
2. Generalized averages. The observation that Cesaro averages may be replaced
by more general averages in the argument above leads to some interesting related
results.
We condider the family Q of all matrices Q — (qUtV) (u, veV) with quv ^ 0,
Συ^Μ,ι; = 1 f°r aH u and
(3.1) lim LI?M-?M+e(ol = °
м-> oo
for i = 1,..., d. It is easy to see that lim quv = 0 uniformly in v.
м->оо
The same arguments as above yield:
Theorem3.5'. Under the assumptions of theorem 3.5 the averages Σν<1ηνΤνχ
formed with any Q e Q converge weakly to ρ for и -> oo.
An equivalent formulation may be given with the help of Lorentz's concept of
almost convergence. A generalized sequence (xu) in 9) is called almost convergent
Ί{Ιίτηπ(η)~ίΣν<ηχν+\ν — x uniformly in we V.
Lemma 3.6. #Σν<1ϋ,νχν ~+ x for all Qe Ώ, then (xv) is almost convergent to x.
This holds for the weak topology as well.
Proof. If (xv) is not almost convergent to χ there exists a neighborhood U of x,
a sequence u{\) <u{2) < ... -* oo and elements w(k)e V with
π(Η(Κ))~1Σν<η(Η)Χν+ι^(^φυ. For и = u(k) put qUtV = п(и)~х if w(k)uv<
и + w(k) and = 0 elsewhere. If и does not belong to the sequence put quv
= %{u)~x for g [0, u\_ and = 0 elsewhere. (It is not hard to see that the almost
convergence of (xv) to χ conversely implies Σ<7ΜΛ -* * f°r aU Q) n

292 Special topics
Lemma 3.7. If(xu) is almost convergent to χ andxu+e{i) — xu converges to Ofor all
i, then xu converges to x.
Proof. We may assume χ = 0. Let U be a neighborhood of 0. Find a convex
neighborhood Ux of 0 with Ux + Ut с U. For some large и е V the averages
y(u, w) = ^w^Xt^M^ + wbelongto Ut for all w. Let C/2 be a neighborhood of 0
which is so small that any sum of at most σ{μ) = ux + u2 + ... + ud vectors in U2
belongs to Ux. Call v9 v' e V neighbors ifv' = v± e(i) for some i. Any two
elements in [w, w + w [ may be connected by a string of at most σ (и) = ux + w2 + · · ·
+ ud neighbors. If w is so large that xv — xv> e U2 holds for any two neighbors v9 v'
in [w, w + w[ and if t; e [w, w + «[, then xw — χυ belongs to Ux.
The identity xw = y(u9 w) — %{u)~l I»6[w,w+«[(^ — ^) an(^ the convexity of
иг now implies xwe U. α
Combining the lemmas with theorem 3.5r we obtain a multiparameter extension
of a result of Bruck [1978]:
Theorem 3.8. Under the assumptions of theorem 3.5, Tux converges weakly iff
Tux — Tu+e(i)x converges weakly to 0 for i = 1,..., d.
3. Strong convergence. It follows from an example of Genel and Lindenstrauss
[1975] that there exists a nonexpansive mapping in the unit ball С of (2 and an
n-l
x0e С for which л-1 X Tkx0 does not converge strongly. Thus strong conver-
fc = 0
gence can only be obtained under supplementary conditions.
Again С is a closed convex subset of a Hubert space §. A semigroup У of
nonexpansive mappings Tu: С -* С is said to be asymptotically isometric on
C0 c: C, if for all x,ye C0, lim,,^^ || Tux — Τ^+^^ΙΙ exists uniformly in ν e V. The
proof that Tux is strongly almost convergent for χ e C0 shall follow easily from
the following Hubert space theorem, which extends a result of Bruck [1978] to
the multiparameter situation and requires a different proof.
Theorem 3.9. If(xU9 ueV) is a generalized sequence in 9) and limu^00 <xM? xu+v}
exists uniformly in veV (and is finite) then (xu) is strongly almost convergent.
If η > 0, we call В cz С η-covered by sets At if the union A of the sets At is
contained in В and card (/l)/card (B) ^ 1 — η. The proof of the following lemma is
elementary and is left to the reader:
Lemma 3.10. Let {v'9 v") be a two point subset ofV9 then given η > 0 we can find
u* e V such that any [w, w + и [ with u^u* can be η-covered by disjoint translates
{v' + v,v" + v}of{v\v").

Nonlinear operators 293
Proof of theorem 3.9. The assumption implies the convergence of ΙΙΣι^Λ+ηΙΙ
to a finite limit for any (<xv) with Σι;Ι°ϋ < °°· Ри* У(г> и) = π (г)-1 Σι;6[ιι,ιι+γ[*»
for г, weV, and /? = liminf^^ lim^^ \\y(r9 u)\\. Then we have Ο^β
<; supiH^H: fe V}=:M<oo.
We treat the case 0 < β < oo, the case β = 0 being similar and simpler.
Multiplying the xv by /?-1 we may assume /? = 1.
Let ε > 0 be given. By the uniform convexity of the norm there exists δ > 0
so that 1 -2<5 ^ ||ζ£|| ^ 1 + 2<5, (i = 1,2), and ||ζ1-ζ2||^ε implies
\\(zi — z2)/2\\ < 1 — 2<5. We may assume δ < ε/2.
Fix r g V with 1 — 5 < limj|>>(r, и)|| < 1 + δ and then fix и* such that
I — <5 < || j;(r, w)|| < 1 + <5holdsforw ^ w^ Ifw^ is large enough the assumption in
the theorem yields the following: if (<xv) is a vector having at most In if) non zero
.components and satisfying Σ | ocv | ^ 2, then the length of any two vectors
Σνοίνχν+Η and ΣυαΛ+Μ' with w, w' ^ и* differs by at most δ.
Let us write rov = (rxt;1? γ2ι;2? ..., rdvd). Assume there exist t/, t/'e V with
t/ °r,u"or^ w*5 and || y(r, t/ ° r) — y(r, υ" ° r) || > 2ε. Given any η > 0, the lemma
implies the existence of w* e V such that any [w, w + w[ with u^ u* can be ?/-
covered with disjoint translates of {v\ v"}. If w* is large enough any [w, w + w[
with w ^ wr := w* ° r is 2f/-covered by disjoint translates
Av = [t/ ° r + t;, t/ ° r + t; + r[υ [ι?" °r + u,u"°r + i; + r[
of A0. From the choice of ub and t/ °r,i)"°r^ и* we obtain the following:
The vectors y(r,v'°r + v) differ in length by less than <5, and also the vectors
y(r,v' °r + v) — y(r, v" °r + v) differ in length by less than <5. Hence
1 - 2(5 < || j>(r, v' о г + v)\\, \\y(r, v'Or + v)\\<l + 2<5,
and \\y{r, v'°r + v) — y{r, v"°r + v)\\ > ε, and therefore
||(y(r, v'or + v) + y(r, vf,or + v))/2|| < 1 - 2δ.
Now у (w, w) is the average of the vectors xt with / e [w, w + и [ and the vector
(>>(r, v' ° r + v) + y(r9 v" ° r + v))/2 is the average over the xt with ίe Av. As the
norms of all xt are bounded by Μ < oo we can achieve \\y(u, w)\\ < 1 — δ for
large enough w and all w. This contradicts β = 1. We have proved the inequality
||j>(r, t/ ° r) — >>(r, t/' or) || ^ 2ε for ι/ °r,v" °r ^ гА
The remainder of the proof is easy. For large и the interval [0, и [ may be
covered, up to a small portion, by disjoint intervals [v°r,v°r + r[ with y°r
^ w*5, and the average over each of these intervals is within 2ε of>>(r, t/ ° r). Hence
II .Km, 0) — y{r, v'°r)\\ < 3ε for large и and \\y(u\ 0) — >>(w, 0)|| < 6ε for large
w, w'. As ε was arbitrary, y{u, 0) converges to some y.
Clearly \\y{r, ν ° r) — y\\ < 4ε for ν ° r ^ u3. For large enough w any interval
[w, w + и [ can be covered arbitrarily well by intervals [vor,v°r + r[ with ι? ° r
^ w^. Hence ||^(«, w) — y\\ < 5ε for large u. As w was arbitrary the convergence
of y(u, w) to у is uniform in w. α

294 Special topics
The desired result on strong almost convergence is now an easy consequence:
Theorem 3.11. IfSf as in theorem 3.5 is asymptotically isometric in {x}, then Tux
is strongly almost convergent.
Proof. Put xu = Tux—p with ρ e F. Then || xu || is decreasing on V and converges.
The assumption implies that limM \\xu+w — xu\\2 exists uniformly in w. Therefore
<xM, xM+w> converges uniformly in w. By theorem 3.9, xu is strongly almost
convergent, α
Odd T. Let us mention at least one condition implying that 5^ is asymptotically
isometric in С Τ is called odd if С = - С and T(- x) = - Tx for χ e С. У is
called odd if all Tu are odd. In this case we find || Tux + Tuy\\2 = \\Tux- Tu(-y)\\2
й \\x + >ΊΙ2· Because of 7Ό = 0, || Tux\\ is a decreasing function of u. The
inequalities \\Tu+0x± Tu+V+Wy\\2 й \\Tux±Tu+wy\\2 yield
2\(Tux,Tu+wy} - (Tu+Vx, Tu+V+Wy} I
^(||Гмх||2-||Гм+|;х||2) + (||Гм+м;^||2-||Гм+|;+м;^||2).
As || Tuχ ||2 and || Tuy\\2 are decreasing functions of и both summands are < ε for
large enough и and all v, w. We obtain
0 й II Tux - Tu+Wy\\2 - || Tu + Vx - Tu+V+Wy\\2
^(||ΓΜχ||2-||ΓΜ + υχ||2) + (||ΓΜ+^||2-||ΓΜ + υ+^||2) + 2β<4β
for large и and all v9 w. This means that \\Tux — Tu+Wy\\ converges uniformly in w.
4. Continuous parameter semigroups. As in the linear case ergodic theorems for
continuous parameter semigroups can be deduced from their discrete
counterparts, but the argument is not the same. We give it only in the one parameter case
although it extends easily to the case d > 1. Let £}R be the family of kernels Q
00
= Ш0, t)\ t,s^0} with Q(s, /)^0 and J Q(s, i)dt = 1, such that (20, ) has
о
finite total variation V{s) for each s, lim^^ V{s) = 0, and
τ
lim J Q{s, i)dt = 0 for all T< oo.
s->oo 0
A semigroup {Tt: t ^ 0} is called strongly continuous, if || Ttx — Ttox\\ -* 0,
(/ -> /0), holds for all /0 = 0 and all x. Then we can define the averages A® x
00
= § Q(s, t)Ttxdt by approximation with Riemann sums. For Q{s,t)
°_ s
= s-1 l[o,s[(0 we obtain the usual averages s~l \Ttxdt.

Nonlinear operators 295
Theorem 3.12. Let С be a closed convex subset of a Hilbert space §, and
{Tt, t ^ 0} a strongly continuous semigroup of nonexpansive mappings Tt: С -> С.
Assume F*= {peC:Ttp=p for all t ^ 0} φ 0. Then, for QeQu, xeC, A$x
converges weakly as s -> oo.
Preo/. We follow Reich [1977]. Consider any sequence sn -* oo. For A > 0 put
(fc + l)ft
1nk = <lnk(h)= J Q(sn9t)dt.
kh
oo
Clearly #nk ^> 0 and Σ 9ιλ = 1 · Thus the estimate
fc = 0
Σ I*m+i -*-..*! ^ Ϊ IGfo.> ^ + A)- β to» 01 ^ Ακω
fc = 0 0
shows (#nk) g Q. Applying theorem 3.5' to the maps 7^(Л) = 7^,, (к Ξ> 0), we find
00
that ^fc)x:= Σ <1пкТк{Н)х converges weakly to some q{h)e C.
fc = 0
Put α (A) = sup {||χ — Ttx\\: 0 rg ί ^ A}. Then it is simple to check
\\А^х-А^пх\\йФ)- This implies \\q(hx) - q(h2)\\ й 0L(ht) + α(Λ2). As a (A)
tends to 0 for A -^ 0 we see that ^:=Итл_0^(А) exists. Now #:= w-lim^n.x;
follows from A%x -q = (Αγηχ - A^x) + (A™x - q{h)) + (q(h) - q). As the
sequence (sn) was arbitrary, the proof is complete, α
The same argument yields an extension of theorem 3.11.
Notes
Simple proofs of theorem 3.5 and 3.5' (for d = 1) have been given by Brezis and Browder
[1976], Pazy [1977], and Bruck [1978]. The limit can be identified as the asymptotic
center of (Tux). (The asymptotic center of (xn) is the element y* of § minimizing F(y)
^limsup^JK-jH).
The weak almost convergence was observed by Bruck [1978] and Reich [1978]. Apart
from their intrinsic interest, the main motivation of nonlinear ergodic theorems so far
seems to be their application in the study of the asymptotic behaviour of solutions of
nonlinear evolution equations, see e.g. Reich [1977]. The multiparameter extension is a
special case of the results of Brezis and Browder [1977], dealing with even more general
semigroups and more general Banach spaces.
The strong convergence of Anx = n'1 (T° + ... + Тп~х)х for odd Τ is due to Baillon
[1976], [1976 a]. Brezis-Browder [1976], [1977] extended the result, and Bruck [1978]
deduced strong convergence for asymptotically isometric Γ and gave quite a few sufficient
conditions for Γ to be asymptotically isometric. Pazy [1977] proved the strong
convergence for nonexpansive Τ with F Φ 0 such that (/ — T) maps bounded closed sets into
closed sets. (An omitted argument has been added in the lecture notes of Pazy [1980]).
Hirano and Takahashi [1979] relaxed the condition of nonexpansiveness of Γ and proved
the weak convergence of An χ assuming 11 Τ1 χ — Tly\\ ^ аь\\х — y\\ witha^ -> 1.They also
showed that the weak convergence of Anx for all χ implies F ή= 0. Takahashi [1981]
proved a nonlinear ergodic theorem for amenable semigroups.

296 Special topics
If С is a bounded closed convex subset of a Banach space X and T: С -> С nonexpan-
sive, then Τ need not have a fixed point. E.g. one can take X = C([0,1]), and С =
{/€Ϊ:0=/(0)^/(ω)^/(1) = 1 for cog [0,1]}, and (Tf) (ω) = ω ·/(ω). Browder
[1965], Kirk [1965], and Gohde [1965] proved the existence of a fixed point for
uniformly convex X.
If Γ has a fixed point it is not certain that Anx converges to a fixed point, even if X has
finite dimension. Sine gave the following example: X = U3 with norm ||(я, ft, c)|| =
Max ((a2 + b2)1/2, | с |), С = closed unit ball and Τ (a, b, c) .= (- b, a, {a2 + b2)l/2). It can
be checked that (0, 0, 0) is the only fixed point and An(a, b, c) -> (0, 0, a2 + b2).
A Banach space X is said to have a Frechet differentiable norm if {\\x — гу\\ — ||χ||)/ε
converges uniformly in χ with ||x|| <Ξ 1 for all y, as ε -> 0.
Reich [1979] proved
Theorem 3.13. Let С be a closed convex subset of a uniformly convex Banach space X which
has a Frechet differentiable norm. For nonexpansive T: С -> С the following conditions are
equivalent:
(a) F+0;
(b) {Tnx,n^0} is bounded for each xeX;
(c) AnTlx converges weakly to some yeC uniformly in i = 0,1,...
A new proof of this theorem has been given by Hirano [1980]. Baillon [1978] proved the
weak convergence of Anx for X = Lp, (1 < ρ < oo). Brezis and Browder [1977] considered
semigroups and general averages under supplementary conditions on X. Bruck [1979] has
simplified Baillons proof for X = Lp and extended it to uniformly rotund X. Reich [1983]
proved a continuous time nonlinear ergodic theorem in Banach spaces assuming only
strong measurability of the semigroup; see also Pazy [1979].
Rode [1982] considers a semigroup G with asymptotically invariant mean {μα} and the
averages Τμα of a semigroup {Tg, ge G} of nonexpansive mappings in Hubert space.
Hirano [1982] has a nonlinear ergodic theorem in which the condition of Frechet
differentiability of the norm is replaced by Opial's condition: w-lim л;п = x0 implies
liminf \\xn — jc0|| < liminf ||xB — x\\ for all χ φ χ0.
Call an operator Τ in an ordered Banach space convex if T(ocx + (1 — a) y) 5Ξ α Τ χ
+ (l-a)7> holds for all x,y and 0 ^ α ^ 1. Rode [1983] showed: if Τ is
nonexpansive, convex and monotonic and if {n~l Tn0, η ^ 1} is relatively compact, then
limw"1 Γ"0 exists.
Djafari-Rouhani and Kakutani [1984] have recently supplied an elegant short proof of
the following result containing Baillon's ergodic theorems: Let (xn) be a bounded sequence
in Hilbert space and an = n~1(x1 + ... + xn). If ||xi+1 — xj+1\\ й ll*i — *;ll holds for i,j
= 0,1, 2,... then (an) converges weakly. If, in addition, \\xi+1 + xj+1 \\ ^ \\xt + Xj\\ holds
for ij = 0,1,... then an converges in norm.
Pazy [1979] and Baillon [1981] have written expository articles and Takahashi [1980]
has a survey on fixed points. Further related references are Bruck [1981], Baillon-Bruck-
Reich [1978], Kobayasi and Miyadera [1980], Miyadera [1978/79], Pazy [1978], and
Gutman-Pazy [1983].

Miscellanea 297
§9.4 Miscellanea
Here we list some further topics related to our main theme. Attempting to limit
the size of the book we just provide a guide to the literature.
1. Random sets. Let d ^ 1 be a fixed integer and let (£ be the family of non empty
compact subsets of Ε — Rd. со (£ denotes the family of non empty convex
compact subsets of 2?, || · || the Euclidean norm, and Bx the closed unit ball in E. On (£
the Hausdorjf metric ρ is defined by
q(C, D) = inf {ε > 0: С с D + eB1 and DcC + вВг}.
We also write ||C|| = sup {||c||: ce C}. A process X = {Xuk: 0 S i < к <оо} of
(£-valued random variables is called subadditive iff i<k<l implies
X{ χ(ω) d ΧίΛ(ω) + Xk,i(u>) everywhere. Ζ is called stationary if the joint
distribution of Zis that of X' with X[ k = Jfi+1,fc+1. The case of sums is the case XUk
fc-1
= Σ Yj where (Y0, Yi9...) is a, stationary sequence of (£-valued random vari-
j = i
ables. Schurger [1983] proved: If Ζ is a stationary and subadditive process of
со K-valued random variables with Ε(\\Χ0Λ 11) < oo, then η_1 Z0 n converges a.e.
in the Hausdorff metric.
Artstein and Vitale [1975] proved this theorem in the case of i.i.d. sums even
for (£-valued random variables. They showed that for sums the (£-valued case can
be reduced to the со K-valued case. Schurger showed that there exist subadditive
CL-valued processes for which n~lX0n fails to converge. The case of general
stationary sums is due to Hess [1979], who considered also generalizations in
real separable Frechet spaces. Schurger [1983], [1984] has also "superstation-
ary" and "almost subadditive" generalizations.
2. Empirical distribution functions. The Glivenko-Cantelli theorem holds, of
course, also for ergodic real valued stationary processes XUX29.... One just has
to replace the strong law by the Birkhoff theorem in the proof. The assumption of
ergodicity can be eliminated if one takes the conditional distribution function of
Xu given the σ-algebra J of invariant events, for the limit. Let Fn{x, ω)
η
= n~x Σ 1 [- «.χ] №(ω)) be the value of the empirical distribution function at χ
ί = 1
after η observations. Tucker [1959] proved: If Xl9 X2,... is stationary, then
sup | Fn{x9 ω) - E{\ t_ ^x] oXx\J) (ω) |
— oo <x< oo
tends to 0 a.e.. Stute and Schumann [1980] extended this to stationary processes
with values in Rd, admitting also other uniformity classes than the class of
intervals [— oo,x] with xe Rd.

298 Special topics
3. Martingales and ergodic theorems. There are rather striking similarities
between the proofs of pointwise ergodic theorems and martingale theorems. This
has led to various attempts to unify martingale theory and ergodic theorems. For
a survey see M.M. Rao [1973].
Jerison [1959] represented ergodic averages An{x)f'm a probability space via
semi-integrable martingales in a suitable σ-finite measure space.
Rota [1961] obtained a generalization of the martingale theorem and the
ergodic theorem for measure preserving flows {τί? / ^ 0} and/e Lp, (1 < ρ < oo),
by considering Reynolds operators R. These are continuous linear operators
in Lp with R(fg) = (Rf)(Rg) + Rl(f-Rf)(g-RgK for f,ge LpnL„. He
proved: If {Rt: t > 0} is a commuting family of positive contractions in Lx of a
probability space which are Reynolds operators and which satisfy (sRt
— tRs)Rsh = (s — t)RtRs h fors> /andAe L^, then lim^^ ^/exists for/e Lp,
(1 <p < oo). He also showed that there are basically two kinds of Reynolds
operators: conditional expectations and those of the form R = (I — D)~x where
D is the infinitesimal generator of a semiflow. For related work see Rota [1964],
Bray [1969].
The above two approaches are restricted to measure preserving
transformations. A. and С Ionescu Tulcea [1963] have proved a maximal inequality
(Theorem 4.2.6) which allows to deduce the ergodic theorem for Lx — L^-
contractions and the martingale theorem even in the vector valued case.
Jacobs [1960] and Neveu [1965 a] have given proofs of the martingale
theorem using the method of proof of ergodic theorems, and Neveu showed that
the decreasing martingale theorem can be considered as a limit case of an ergodic
theorem of A. and С Ionescu Tulcea.
4. Non homogeneous Markov chains. As the concept of an "ergodic theorem"
proposed in the introduction assumes the existence of a semigroup of
transformations, the theory of non homogeneous chains need not be treated here. The
books of Seneta [1973] and Isaacson-Madsen [1976] treat this topic and give
references. The papers of Cohn [1977], [1979] and Taylor [1978] are more
recent references which came to my attention. Jacobs [1957a], [1958], [1960]
has studied Markov processes with almost periodic transition probabilities.
5. Ergodic theorems in demography. These theorems deal with asymptotic
properties of products of non negative matrices. A survey has been given by J. Cohen
[1979]. In the basic model one considers a column vector Y{t) with non negative
components Yi(t), (i = 1,..., k), representing the number of females in age class i
at time ieZ+. One assumes Y(t + I) = x(t + 1) Υ if), where x{t) is a non
negative matrix with xM = 0 for i ^ 2, / ^ i. The "strong ergodic theorem" assumes
x(t) constant and studies the population size || Y{t)\\ = | Yt (t) | + ... + | Yk(t) \ and
the age distribution Γ(/)/ΙΙ 54011 wm the Perron-Frobenius theory. The "weak
ergodic theorem" assumes a deterministic inhomogeneous sequence x(t) and

Miscellanea 299
shows how the age distribution of different initial populations becomes more and
more similar. There are also models where the x{t) are chosen at random. For the
Markov chain case see Lange [1979].
Brillinger [1981] has surveyed tools and models in population mathematics.
6. Nonlinear averages. Convergence problems for one-dimensional infinite
particle systems motivated work by Fichtner-Freudenberg [1979] on nonlinear
averages:
Let Y= (Y^iei be a stationary ergodic sequence of non negative random
variables with finite expectation and let (/„) be a sequence of functions mapping
Ω = Еж (with Ε = [0, oo]) into R. One is interested in conditions on (/„) implying
fn(Y) -> EY0 in Lx or at least in the weak topology. The Cesaro averages
correspond to/„(>>) = n~x (y0 + yx + ... + Уп-ι)' But there is the need to consider
nonlinear functions.
Let M be the set of all probability measures μ which are invariant under the
shift θ and satisfy μ (] 0, oo [z) = 1 and J X0 άμ < oo. (Xj (ω) = α^ is they-th
coordinate of ω). Let ^ be the set of measurable maps/: Ω -* Ε which are non
decreasing in each coordinate and satisfy \/άμ = J X0 άμ for all μ e M. E.g., the function
/* with/* (ω) = lim sup л-1 (ω0 + ...+ω„_1) belongs to #\ If h is differentiable
with 0 й dh{t)\dt <; 1, then fh(ω) = ω0 + h((ox) - Α(ω0) belongs to J*\
Call (/„) с 3F weakly asymptotically 0-invariant if w-lim (/„ — fn ° θ) = 0 holds
in Lx (μ) for all μβ Ж. Fichtner-Freudenberg show that this is equivalent to
w-limfn = /* in Σχ(μ) for all μ^Μ. They give sufficient conditions for the
norm-convergence in Li (μ) of/„ to/*.

Supplement
Harris Processes, Special Functions,
Zero-two law
Antoine Brunei
This supplement deals with an important class of quasi-compact operators
arising in the theory of Markov processes.
1. Taboo potentials, integral kernels. Let (Ω, sf9 μ) be a σ-finite measure space,
and let Ρ: (ω, Α) -> Ρ (ω, A) be a stochastic kernel. We also denote by Ρ the
operator in the space bs/ of bounded ^-measurable functions given by formula
(3.1.1). Η denotes the space of ^-measurable functions h: Ω -* [0,1] with
μ (h) := J }ιάμ > 0. Ih shall be the operator of multiplication by h.
Definition 1. The taboo potential Uh associated with he His the operator given
by
00 00
uh= Σ {Ph-h)nP= Σ Pih-hPf,
n=0 n=0
where (71_ΛΡ)° = /.
Note that C4/is well defined as a measurable function with values in [0, oo] when
/is non negative and measurable. For A e sf9 we put UA= UlA.
The Markov process associated with the stochastic kernel Ρ is called recurrent
in the sense of Harris if the following condition (H') is satisfied:
(Hr) UAlA = i for all A e si with μ(Α)>0.
As (ΡΙΑο)ηΡ1Α(ω) is the probability that the first visit to A9 given a start in ω,
happens at time η + 1, the condition (H') means that the probability of a visit in A
at some time m ^ 1, given a start in ω, is 1 for all сое Ω and all A having positive
measure.
The taboo potentials Uh satisfy a fundamental relation, called the resolvent
equation:
Proposition 2. For h,ke Η with h ^ к the following equalities hold
(R) Uh=Uk+ UhIk-h Uk=Uk+ UkIk_h Uh.

302 Supplement: Harris processes
Proof. Let us write Ix _ЙР = X, Ix _ki> = Υ and Χ- Υ = Ik_hP = Z. We denote
00
the formal series Σ f = 1 + / + t2 + ... by C?(/).
n = 0
Then we can write
Uh = P Σ (Υ + Ζ)" = i>G(7 + Z)? Uk = PG(Y),
n = 0
and it suffices to verify the identity
G(Y + Z) = G(Y) + С?(Г+ Z) ZC?(r)?
in which У and Ζ are positive operators. For this it is enough to verify the identity
G(u + v) = G(u) + G(u + v)vG(u)
for formal series with non commuting variables u, v. Starting from the relations
G(u + υ) (1 - и - υ) = 1 = (1 - и) G(u)
it suffices to write
1 — и = (1 — и — ν) + ν,
and to multiply this identity from the left with G(u + v) and from the right with
G(u). α
We now derive some immediate consequences of the resolvent equation. The
identity Uhh = H implies Ukk = H for all к e Η with к ^ h. Indeed, Ukh й Uhh
= H and (R) yield
1 = Uhh = J7kA + UkIk.h Uhh = J7kA + C4(fc - A) = f/kfc.
For A with μ(/1) > 0 and 0 < θ < 1 the resolvent equation implies
υ$ΐΛιΑ = υΛιΑ + υ$ΐΛ{ΐ-θ)ΐΐΛυΛιΛ9
and if L^l^ = 1, it follows that
UeiAiA = l+Q-0)U$lAlA.
This leads to Ue x A (Θ1A) = H. Now, (or he Η there exist >4 e л/ with μ (yl) > 0 and
θ with 0 < θ < 1 and Θ1Α^ h. The preceding argument then proves £/йА = 1.
Thus the condition
(H) C4A = 1 for all he Η
is equivalent to (#') and we assume in the sequel that it is satisfied.
The definitions and constructions just given for a kernel extend easily to the case
where Ρ is the adjoint Γ* of a positive contraction Tin Lx (Ω, sf, μ). Thus Ρ acts
in L^ (Ω, s/9 μ). Let us add the following remark: A kernel Ρ acts on the space of

Supplement: Harris processes 303
bounded measures by the formula ν Ρ (A) = §P(x, A) dv(x), and for/e hstf one
has
(vP,f} = (v,Pf} = $Pf(x)dv(x);
in the case Ρ = Γ* we can similarly regard Ρ as an operator acting from the left
on measures ν of the form ν = gμ with ge Lt such that ν Ρ = {Tg)μ and
<vP,/> = <7fc,/> = <g, Γ*/> = <v? P/> = lPf(x)g(x) άμ{χ).
With these conventions the following material can be read either for kernels, or
for operators when the bounded measures are replaced by Lx · μ and the space
bstf by L^. It is also clear that proposition 2 applies to both kinds of operators as
its proof was purely formal.
Taking h = 0 in (R) we obtain
U0=Uk+U0IkUk
and hence
U0k = i + U0k.
This yields U0k = + oo for all к e H. If Ρ is the adjoint of a contraction Γ in L1?
the last relation shows that Τ is conservative and ergodic.
For Ae si with μ (A) > 0 the operator IA UAIA is the operator induced by Ρ on A.
It is also useful to consider the balayage operator
00
n = 0
and to put RA = R1a. For given ^4 and for Ecu A with //(i?) > 0 we put
00
Г = /<ι/,/, and ϋ'Ε = ΙΑΣ (Ιλ\βΡΎΙλ·
п = 0
The identity R'E = IAREIA may be verified in a formal way just like the resolvent
equation. Hence, the identity R'E\E = 1A yields U'E1E = P'R'E1E = l^.This
relation expresses the Harris condition for the operator P' acting on b{si η A), resp.
L^ (/1). In particular it shows that the induced operator is again conservative and
ergodic.
It is time now to emphasize an essential difference between the case of an
abstract contraction in Lx and the kernel case. In the latter case the Harris
condition implies the irreducibility of the Markov chain with respect to the
measure μ. In other words one has, for all ω e Ω, the absolute continuity μ<^υ0 (ω, ·);
this condition can also be written as μ <ζ Ue(co,) for all сое Ω and some
0e ]0,1 [. Indeed, if U0(co, A) = U0lA(co) = 0, then one has also UAlA(co) = 0
because of UA rg U0.
On the other hand, we can define the notion of abstract Harris operator for a

304 Supplement: Harris processes
positive contraction in Lx following ideas of Foguel [1979] and Foguel and
Ghoussoub [1979]. To this end we recall a definition:
Definition 3. A positive linear operators K: L^ -> L^ is called integral kernel (for
the density k) if it is of the form
where k:QxQ-+R+ is a positive si ® ^/-measurable function with
supx { Jk{x, γ)άμ(γ)} < oo. We say that К is поп trivial if 7П Φ 0.
In the special case k(x,y) = a(x) b(y) we shall write К— а ® bμ. Then
Using this, the abstract notion will be given by
Definition4. A positive contraction Tin Lx and its adjoint T* = Ρ are called
Harris operators if (#) holds and there exists an«eM such that Pn dominates a
non trivial integral kernel.
(Note, that (H) holds for Ρ if and only if Τ is conservative and ergodic).
We shall need an elementary property of integral kernels expressed by
Lemma 5. Each positive linear operator in L^ dominated by an integral kernel is
itself an integral kernel.
Proof. Let Q and КЫ positive linear operators in L^, and assume that К is an
integral kernel dominating Q. We associate with Q a finitely additive measure Q
on the algebra in Ω χ Ω generated by si χ si. Q is determined by
ζ>(ΑχΒ) = ΙαίΒ(χ)(Ιμ(χ), (Α, Β est).
A
In the same way ^determines K, and clearly Q^K. But £ extends to a measure
on the σ-algebra si (x) si: For Be si (x) si one has
Ζ(Β) = \1ζ(χ9γ)άμ{χ)άμ(γ).
в
Consequently, Q extends to a measure on si ® si, too. As this measure is
absolutely continuous with respect to К it is absolutely continuous with respect to
μ (χ) μ. It has a density dominated by k. α
From this we easily deduce
Proposition 6. Assume 0 < θ < 1. A conservative and ergodic Ρ = T* is a Harris
operator if and only if U0 dominates a non trivial integral kernel.

Supplement: Harris processes 305
Proof. We apply the inequality (А + В)лС^АлС + ВлС, which holds for
positive operators А, В, C, because (А + В)лС^(АлС + В)лС^АлС
+ 5 л С.
It is sufficient to show that when К S Ueis an integral kernel with Pn л К = 0
for all и, then К = 0. Now
Κ=υθΛΚ^(Σ (1-0)пРп+1)лК+( Σ (1-й)"Ри+1)л^
η = 0 η = Ν + 1
00
й( Σ (1-0)"Р"+1)лл:.
η = Ν + 1
Letting Ν -* οο, one obtains ΑΓ= 0. α
2. Special functions and invariant measures. We now introduce the class of special
functions for P.
Definition?. A non negative measurable function/is called special if Uhf
belongs to L^ for all he H. Let S denote the family of special functions. A
set A e si is called special if 1A e S.
It is expedient to begin with the case where Ω is special. Let 0 < θ < 1 be given.
We shall establish that the assumption 1eS yields, for each Ее si with
μ(Ε) > 0, the existence of a constant b(E) > 0 with UelE^b(E).WQ can write
1 = PI = P1E + i>l£c = P\E + (PW1 = P\E + №c) (P1E + (P/£c)1)
= Σ (Р/£сУ>1£ + (Р/£с)21,
and, by recurrence,
1= Σ (PIEcyPlE + (PIEC)n+4.
j = 0
Now the assumption 1eS implies £/EH rg α for some aeU. Hence
00 00
UEi = Σ (PIEc)jP^ = Σ (PIe-)^ й a.
j=0 j=0
As the sequence (PIEc)ni is decreasing, one obtains
||(P/£,)l = ||(P/£c)"1||^^-|0.
η + 1
This yields the uniform convergence to H of the sequence Σ (Р/есУ PI
^Therefore there exist peN and b' > 0 with j=0
Σ ^1ε^ Σ (i>/Ecyi£^'.
j=0 j = 0

306 Supplement: Harris processes
Apparently this implies the existence of b(E) S UelE.We make this more precise
in
Lemma 8. Ifi e S, there exists 0 < ε < 1 such that μ(Β) ^ 1 — ε implies UelB
Proof. Otherwise there exists a sequence (Aj9j = 2, 3,...) with μ(Α^ ^ 1 — 2~j
and U9 1A. < 2 "Jon a set Ц with μ (2$) > 0. The set Ε = f]j^2 ^j tben has measure
;> 1/2 and t/el£ ^ Ue\A. й 2~j on 4.. This contradicts 0 < b(E) ^ U01E. и
An important consequence of the preceding result is
Theorem 9. (Horowitz) If Ω is special, then Ρ is a Harris operator.
Proof Let S be the operator Ι ® μ. From § 4.1 we have
(Ue л S)1 = inf {(t/,g+ J(1 -g№): 0 £ g £ Ц.
Fix g and put В = {g ^ 1/2}. Then
If μ(Β£) > ε, then Ueg + J(1 - g)<//i ^ e/2. If μ(Β<) й β, then μ(Β) ^ 1 - ε and
lemma 8 implies U9 1B ^ ε. It follows that Ue л Sis not trivial, and, hence, that Ρ
is Harris, α
There is another useful property of integral kernels. Following Foguel [1979] we
establish
Lemma 10. Let Q be the adjoint of a positive contraction in Lx and Kan integral
kernel. Then QK and KQ are integral kernels.
Proof. The result is clear if AT is the kernel K0 = H (χ) μ. Another easy case is that
where the density k(.,.) is bounded by some M, because then К rg MK0 and the
inequalities QK^ MQK0, KQ <; M(K0Q), together with lemma 5, imply the
desired property. In the general case it is enough to truncate the unbounded
density putting kn — k/\n. Then the associated integral kernels Kn increase and
they tend to K, and the result follows from QKn | QK and KnQ \ KQ. π
We now apply this to the operator Ue. Putting Θ' = |/0, the resolvent equation
shows Ue ^ (0' — Θ) UQ> Up. If Ue dominates the non trivial integral kernel K, one
can infer that
υ$*(θ' -θ)Κυ$. = (θ' -Θ)Κ19

Supplement: Harris processes 307
where Kx is an integral kernel having a density kx which is strictly positive μ® μ
almost everywhere. Indeed, if μ (Β) > 0, lemma 8 yields
κίιΒ = κυθΛΒ^κβ>ο.
Changing kx on a μ ® μ-nullset we can assume that kx is strictly positive
everywhere.
A lemma of Harris [1956] now implies that the density g of the kernel K\
satisfies an inequality q{x,y) ^ a(x)b(y), where a, b are two strictly positive
measurable functions on Ω. In fact, this can be verified as follows: For any integer
/ > 0 the function
χ -* Мг(х) = μ({ζ: kx (χ, ζ) ^ 1//})
is measurable and it satisfies lim^^Mt(x) = 1. Let l(x) be the smallest integer
with Mt(x) ^ 3/4. Put a{x) = /(x)/2. Then α(·) is strictly positive and the
measurable set Ax = {k1(x, ·) ^ a{x)} has measure μ{Αχ) ^ 3/4. In the same
way there exists a measurable function b(-)>0 such that
By = {кх(-,у) ^ b{y)} has measure μ{Βγ) ^ 3/4. Now
q(x,y) ^ J k1(x,z)k1(z,y)μ(dz) ^ 2α(χ)6(>>)μ(Λ^Λ) ^ α(χ)Ζ>(>0.
Now £// ^ (0' — 0)2 ATf proves that U92 dominates an integral kernel of the form
a ® bμ. As this is true for all θ with 0 < θ < 1 there exist two strictly positive
measurable functions A, g with U^^h® gμ. Consequently
Ue ^ (Θ' - Θ) Ue, (h ® gμ) = (0' - θ) (Ue,h) ® gμ.
Making use of the assumption 1 e S, we deduce V9. h ^ β > 0. This yields
Proposition 11. If Ω is special there exists for all θ with 0 < θ < 1 a measure μθ ~ μ
with Ue ^ H ® μθ.
Corollary 12. If Ω is special there exists a unique invariant probability measure
λ — λΡ equivalent to μ.
Proof The operator Q — θ Ue is Markovian, and the measure ν = θμθ is
equivalent to μ. The inequality Q ^ H ® ν implies Qn^\ ® ν for all η e N. It follows
that
Ιιτηιηΐ:$ζ)η1Αάμ^ν(Α)>0
η
holds for all A with μ (A) > 0. Theorem 3.4.2 yields the existence of a
probability measure λ ~ μ with XQ = A. Applying the resolvent equation to U9 and to
Щ = Ρ we obtain
υ$ = Ρ+υ$(ί-θ)Ρ, Q = eP + (l-e)QP,

308 Supplement: Harris processes
which implies λ = θλΡ + (1 — θ)λΡ = λΡ. The uniqueness is a consequence of
the ergodicity. α
We now turn to the more general situation where there exists a special set A but Ω
need not be special. Let A e si be a fixed set with l^eS.
Recall that the operator induced by Ρ on A is P'= IAUAIA, and that the
balayage operator associated with a measurable Ε cz A is the operator R'E
= IAReIa.
Proposition 13. If A e si is special, the induced operator P' is Harris on A.
Proof. It follows from Щ = P'R'E and IAR'EIA = IAREIA, that UE\A belongs to
L^iA). The resolvent equation shows that this remains true for Uq1e\a for all θ
with 0 < θ < 1. Hence Щ\А belongs to L^A) for all A with 0 ^ A ^ 1л and
μ (A) > 0. It remains to apply theorem 9. α
It is now natural to ask the following question: If Ρ is Harris, do there exist
special sets, and does there exist an invariant measure? Indeed, we have:
Theorem 14. If the operator Τ (or P) is a Harris operator, there exist поп zero
special functions, and there exists a σ-finite invariant measure equivalent to μ. This
measure is unique up to a constant positive factor.
Proof. 1) The construction explained just before proposition 11 shows that, for
given θ with 0 < θ < 1, there exist two strictly positive measurable functions A, g
β
with Ue^h® gμ. If we put Ax = А л -, then equation (R) gives
Uki ^ Uhl(Ie_hi)Ue г\икхие^в- Цк1(А1 ® gμ).
Hence Uhl ^ 11 ® μΐ9 where μχ ~ μ and Ax e Η is strictly positive. We shall now
show that hi is special, and, more generally, that each non negative measurable
function/with Uhlfe L^ is special. Let/^ 0 with UhlfS a<co and h e Я with
h й Αι be given. Applying (R) and using (UhlIhl^h)i = Uhl(h1-h) = 11 - Uhlh
^ H — μ1 (A) we obtain
uj= Σ (uhlihl-,yuhlfua Σ (^Λ-*)"1
йа Σ (1-μ1(Α))" = ο/μ1(Α)<οο.
и = 0
Thus i/A/is bounded for A e Я with A iS At. If A is arbitrary in #, put A' = А л hx.
Then A' ^ At yields Uhfu Uh.fe Lm.

Supplement: Harris processes 309
2) For any θ > 0 the set A = {hx ^ 0} is a special set because 1A^ h1/6. By
choosing θ small we can assume μ(Α) > 0. The second assertion now follows
from proposition 13, corollary 12 and the following theorem which is of
independent interest, α
Theorem 15. Let P' be the induced operator on a subset Ae si. Assume there
exists a finite measure λΑ with λΑ = λΑΡ' — λΑΙΑ. Then the measure λ — λΑ UA
00
satisfies λΡ = λ. If Σ Ρη 1A is strictly positive everywhere, λ is σ-finite. If Ρ = Τ*
л = 0
satisfies (Η), μ (Α) > 0 αηάλΑ ~ μΙΑ) then λ~ μ, and any other P-invariant λ' ~ μ
is a constant multiple of λ.
Proof The iMnvariance follows from
χ = χΑ υΑ = λΑρ + λΑ uAiAcP = λΐΑρ + xiAcp = λΡ.
To prove the σ-finiteness of Я first check UAIAi rg 1. Under the stated
assumption, Ω is the union of the sets В = Bkn = {PniA> i/k}9 and we have
λ (Я) й <λ9 кРЧАУ = к(ХР\ ίΑ} = к (λ, 1А}
= к^АиА91АЛУйк<ХА9иА1АЛУйкХА(0)<со.
In the second assertion λ <ζ μ follows from Ρ = Τ* and λΑ <ζ μ. The ergodicity of
Γ and μ (Α) > 0 imply that the Γ-invariant function/ = άλ\άμ is positive a.e.. If/7
is another non negative ^-invariant (not necessarily integrable) function, then
fAf,==g is sub-invariant. By lemma 3.3.10 and Lemma 3.3.11, {g>0} is
Γ-absorbing. Hence {g > 0} = 0 or = Ω. If there exists a σ-finite invariant
measure Я* which is not a constant multiple of Я, then the density /' of a suitable
multiple of Я* has the property 0 Φ {g > 0} Φ Ω. α
3. Zero-two laws and ergodic theorems. We now establish a result due to Foguel
[1979]; see also Foguel-Ghoussoub [1979]. It will lead us to the ergodic theorem
for Harris operators. A variant will give us the celebrated zero-two law of
Ornstein-Sucheston [1970].
Lemma 16. Let P.Qi.Qi be commuting positive linear operators in L^ with
Pi = Qx\ = Q2\ — \.If there exist an integer rand a positive operator R with ΒΛ
= δ>09Ρ'^ Q.Rand Pr ^ Q2R9 then \imm\\{Q2 - Qx)Pm\\ = 0.
Proof As the sequence ||(Q2 — Qi)Pm\\ is decreasing we may assume r = 1. Put
Q = 021 + бг)/2. We can write Ρ = QR + Sl9 where S1 is positive. Inductively,
we obtain Pn = QnRn + Sn with Sn ^ 0. Indeed, we have
p»+* = p(QnRn + Sn) = QnPRn + PSn
= Qn+iRn+1 + (QnS1Rn + PSn) = Qn+1Rn+1 + Sn+l9

310 Supplement: Harris processes
where Sn+1 = QnSxRn + PSn is a positive operator. It follows from i>wH = H
= QnRni + SJ =δη + SJ, that SJ = 1 - δη.
The second step is the proof of
Pnj = QnTnJ + Si with 7^0
by induction in j. If this has been verified for an integer j ^ 1, we have
PnU+1) = Pn(Qn TnJ + SJ„) = Q"(P" T„fJ + R"Sl) + SJ„+1
= QnT„,J+1 + Si+i
with THJ+1 = P" THj + R"Sl Si 0, ΤηΛ = R\
Put βη] = TnJi. By induction, finjis a constant, because Τ„Λί = Rni = δ". We
obtain
P"'1 = 1 = /?„,,· + Sii = jSBj + (1 - «5"У.
Hence || r„J| = |l 2;,·111 = &,·<!·
We now look for the limit L of the decreasing sequence \\(Q2 — Qi)P"\\· The
inequality
HQ2-Qi)PHi\\uHQ2-Qi)Qn\\ + 2(i-sy
shows L ^ || (g2 - β^βΊΙ since lim (1 - 5"У = О. But
ΙΙ(β2-βι)βΊΙ = ΙΙ(β2-βι)ι'βι + β2'
! ! η -1
< 1— У
^ ^ fc = 0
n\ ί η
Jfc/U + 1
The sequence к -* ( J is at first increasing and then decreasing with a maximum
at (rJ2]Y By Stirling's formula, 2~"f " J = О (1/jA). Hence
Z, ^ lim ||(β2 - β^β-ΙΙ = 0. α
We are now ready to prove
Theorem 17 (Ornstein-Metivier-Brunel). For any special set A for Ρ there exists a
constant MA < oo such that
II Σ rfUuMAfL·
n = 0
holds for all N and for all fe L^iA) with X{f) — 0.
(Recall that λ was a σ-finite invariant measure, and that λ (A) is finite for special
sets A).

Supplement: Harris processes 311
Proof. For given θ e ] 0,1] let U£ denote the taboo potential operator in L^A)
associated with P' = IA UAIA. We know from proposition 11 that there exists a
measure λ' ~ λΐΑ on A with
Щ = Σ (1 - θ)" (ΙΑ UAIA)"+1 ^\Λ®λ' on LX(A).
n = 0
Let/lA = λΙΑ/λ(Α) be the P'-invariant probability measure. It is also invariant
under вЦ. We put
Κ = θ1Α®λ', Ε=1Α®λΑ.
Using the relations
R = ER = eU'eR, 0U'9E = Εθυ'θ = Ε = Ε2
we find that
9U^R = (0US)R, θυ'θ ^ ER.
As U'9 and Ε commute and R\A = βλ'{A) > 0, lemma 16 applies with Ρ = θυ'θ9
Q2 = θϋ'θ, r=i,Q1 = E and yields
lim||(0C/i - Ε) (θυ'θ)η\\ = lim \\(0Щ)Я - E\\ = 0.
£"is the zero operator on the subspace (IA — E) L^ (A). Thus, on this subspace we
have ||(0ί/θ')η|| -► 0. On the other hand, one has
P'-IA = {P-I){IA + IACUAIA) on L„{A).
(Apply (R) to UA and U0 = P).
Now take/e L^C^) with \fdXA = 0, then/e {IA — £") L^C^). Our
considerations show the existence of an integer η (A) for which IA — (0 C/J)"^ is invertible.
We deduce the existence of functions/1?/2?/з е £«,04) and Fe L^ with
/= (iA - №ТМ))Л = (/, - ещ/2 = (/κ - n/з = ('- W
For example F= {IA + IAcUAIA)f3. It is clear that Ц/П1/11/Н is majorized by a
constant which depends only on Α. α
We finish our study of relations between the Harris condition and the existence of
special functions with
Theorem 18. If Τ is conservative and ergodic and there exists a special set A for
Ρ = T* then Τ is a Harris operator.
N
Proof As Ρ is conservative and ergodic and μ (A) > 0 we infer that lim Σ Pn 1A
N n = 0
= oo a.e.. By Egorov's theorem there exists a set В cz A with μ(Β) > 0 on which

312 Supplement: Harris processes
N
the sequence Σ PnlA converges uniformly. Let λΑ be the probability measure
n = 0
which is invariant under the operator P' induced by Ρ on A. We obtain that for
any he Η with h^\A there exists a smallest integer N{h) with the property
N(h)
Σ Ρη\Α^2ΜΑλ-/{Κ)\Β.
n = 0
Applying theorem 17 to the function/= 1A — X^(K)h, which has norm \\/\\ю
й λ^1 (h) we find
sup|| Σ F4\A-k?{h)h)\\uMAk?{h).
N n = 0
This yields
N(h) N(h)
2MAX-A\h)\BS Σ P*\AuMAk?(h) + k?(h) Σ P"h,
n=0 n = 0
and, hence
N(h)
Σ Pnh^MA\B.
n = 0
Now, for 0 < θ < 1, we have
N(h)
Σ
n=0 n=0
(1) Ueh= Σ (1-0)"Ρ"+1Α^ Σ (ί-θ)"Ρη+1Η^(1-θ)Ν^ΜΑ1Β.
The last inequality will allow us to prove that Ue dominates a non trivial integral
kernel. Put К = i ® XA. Recall
(Ue a K)1 = inf {U9g + (J(1 -g)dXA)V.O ugu^}-
Let С = A n {g ^ 1/2}. //AA(C) ^ 1/2, then J(1 - g)dXA ^ 1/2. If XA(C) > 1/2,
then
(2) sup|| Σ Рп(1А-ХА(СуНс)\\йМАХ2ЧС)й2МА.
Ν η = 0
Let 7VX be the minimal integer with Σ Рп^а = ^Ma\b. Nx depends only on A
n = 0
Ni Ni
and B. (2) now yields Σ PnXA1{C)\c^2MA\B. It follows that Σ Png
^MAiB. n = 0 n=0
The minimality of N(g) yields Nt ^ N(g). Using (1) we obtain
Ueg ^ (1 - θ)ΝιΜΑίΒ. Combining both cases we arrive at (Ue л ЛГ)И
^ Min (^ 11,(1 — θ)ΝίΜΑίΒ) ή= 0.ивлК is an integral kernel because it is
dominated by the integral kernel K. Thus Ue dominates a non trivial integral kernel
and Г is a Harris operator, α

Supplement: Harris processes 313
The properties of uniform convergence considered above lead to a
characterization of Harris operators in terms of their quasi-compactness. This notion has
been studied in § 2.2. The relation to present material is made precise by
Proposition 19. For a positive contraction Τ in Lx the following conditions are
equivalent
(i) Ρ = Τ* obeys a uniform ergodic theorem in the form
lim||-WE Pk~^ ®A|| = 0,
η П fc = o
where λ is a P-invariant probability measure equivalent to μ;
(ii) Τ is conservative and ergodic and H is special·,
(Hi) Ρ is quasi-compact, conservative and ergodic.
Proof The key part of the proof is established in § 2.2. The only new point
concerns condition (ii). It is therefore sufficient to prove (ii) => (i) and (iii) => (ii).
(ii) => (i) is an immediate consequence of theorem 17. (iii) => (ii): The condition of
quasi-compactness yields the existence of a sequence (Qn) of compact operators
of L^ into itself with lim || Pn — Qn ||Loo = 0. Let L% be the subspace of L^
consisting of the functions with \fdX = 0 and let π be the operator / — (1 ® λ) mapping
L^ into L°. The inequalities
\\P»-nQn\\Lo„u\\nPn-nQn\\L„
show lim ||i>w - nQn\\L% = 0, and hence, (I-P)LOD = L£.
As in the proof of theorem 17 the statement ||Ρη||^ -* 0 implies that Ρ is a
Harris operator for which Ω is special, α
To complete this study it is of interest to prove, in the abstract case, a lemma
established by J.Neveu [1972] for Markov chains.
Lemma 20. Let Τ be a conservative and ergodic positive contraction in Lv Then
the inequality
tfo 1x^(1-0)™ a.e.
holds for all θ e ] 0,1 [, all A e si with μ{Α) > 0 and all he H.
Proof We first observe that apf й P(af) holds for all a e ] 0,1 [ and 0 uf This
consequence of a (generalized) Jensen inequality for Ρ can be verified as follows:
There exists a countable family (un, vn) e R2 with
d = sup (un t + vn) for all / e R.
η
(It suffices to consider the tangents to the graph of the function t -> d at the

314 Supplement: Harris processes
points with rational /.) If follows that
unf+ vn ^ af
holds (a.e.) for all n. Using Ρ ^ 0 and PH = 1 we obtain
Taking the supremum over η we arrive at apf rg P(af).
It is sufficient for the proof of the lemma to prove Reh\A ^ (1 — 6)RaH, and,
morevover, it is enough to show
1А^(1-1^вкР)((1-вГ*н),
since Reh(I — h-ehP) = I- To this end, it is enough to verify that (1 — 9)f =
(1 _ 0Л)р((1 - θ)1) holds on Ac for /= RAh. But /= A + Pi?^ and the
inequality (1 - 0)* g (1 - 0A) imply
(1 - 0)" (1 - 0)р/^й ^ (1 - 0A) (1 - 0)р/ий й (1 - 0A) P((l - 0)*АЙ) · □
We now arrive at a zero-two law. We follow the proof by Foguel [1971]:
Theorem 21. Let Τ be a Markovian positive contraction in Lu Ρ = Γ*, and
letk>0 be an integer. If \\Pno+k — Pn°\\<2 holds for some n0, then
limj|i>w+k-i>n|| = 0.
Proof Let W= Pno+k a Pno. We have
W\ = inf (Pn°+kg + Pno(i- g)).
Writing g = (1 +/)/2 we find
W\ = inf [i>"0+fc HyM + ^"° \~2)
= inf [1 + X- (Pn°+kf- i>"°/)]
1
^1 --\\Pn°+k-Pn°\\ = d>0.
The function φ = δ/ W\ therefore belongs to H. If R is the operator Ιφ W we
obtain W^R and i?H = δ > 0. Therefore we have the inequalities
pno+k^pkR m =δ>ο, pno+k^/i?
which show that the operators P, Qx — Pk, Q2 — I,R and the integer r = n0 — k
satisfy the assumptions of lemma 16. This yields the desired conclusion, α

Supplement: Harris processes 315
The zero-two law of Ornstein-Sucheston is the deeper "pointwise" form of this
result. It can be formulated as follows:
Theorem 22. If, for η = 1, 2,..., Pn — T*n are conservative and ergodic Markov
operators, then we have either
for all к = 1,2,..Aim{ sup (P»+k/-P»/)} = 0a.e.
or
for all к = 1,2,... sup (Pn+kf- Pnf) = 2 a.e..
Proof. Let к > 0 be a fixed integer and put
hn:= sup (P*+kf-P*f),
and Wn = Pn+k a Pn. As above, we have
WJ=i-hJ2.
In particular, this shows that 0 rg hn rg 2. The properties hn+1 = hn and hn+1
= Phn are immediate. If g ·= lim„ hn, we obtain g = Pg. Hence, g = constant = a
with 0 = a = 2. We have to establish a£]0, 2[.
Assume 0 < a < 2 holds. Then WJ increases to 1 - a/2 > 0. Clearly, Pn ^ Wn,
Pn+k = Wn and Pn+k = WnPk. Hence Pn+k = WnM holds for Μ = (/+ Pk)/2.
If ni9 n2,..., nr is any finite sequence of integers, and m ··= nx + n2 + ... + nr
+ kr, we obtain i>w ^ Wni... ^rMr. We can exploit the fact that
\\(I-P)M'\\ = 0(l/)ft),
and choose r so large that || (/ — Pk) Mr \ < ε/2, where ε > 0 is arbitrarily small. We
can then choose nl9 ...,nr in such a way that, for 7j := Wni·... WHr, one has
7111 φ 0. Now Pm = TxMr + U, where C/is a positive operator. By recurrence iny
we find that Pmj is of the form 7] AT + t/·7', where 7} is a positive operator. It
follows from Pjmi = 1 = 7)1 + C/^'H that 1 ^ C/H, and, hence, that
H ^ C/1 ^ ... Un and 0 Φ 7111 ^ 7} 11.
If A := lim,- С/'И, then h = Uh = Pmh, which yields A = constant. If Α φ 0, then
11 = C/1. But this contradicts to 7)1 φ 0. Hence UH decreases to 0 a.e..
For/e L^ with — l rg/rg H we can write
This yields
hjm = sup (pJm+kf- pJmf) < ε/2 + 2^11.

316 Supplement: Harris processes
We obtain lim sup jhjm = limhn = a rg ε/2. As ε was arbitrarily small, this
contradicts to a > 0. α
4. Eigenvalues and quasi-compactness. To complete the study of the strong
ergodic theorem (Proposition 18), we shall show directly how the spectral
properties of the operator Ρ = T* are related to the quasi-compactness when Ω is a
special set. This proof is based on a lemma about compactness in Lx due to
A. Brunei, and it has been developed by Brunei and Revuz [1974a].
Lemma 23. If(fn) is a sequence which is norm-bounded in Lx of a probability space
(Ω, stf, X)andwhichsatisfieslimnfP^O0\\\fn-fP\ ~ ll/«-/Pllilli = 0, then (/„)
contains a Cauchy sequence.
Proof. The double sequence Unp = J \fn —fp\ is bounded. The two-dimensional
Bolzano-WeierstraB theorem (see Sucheston [1959], lemma 3, p. 390)
therefore yields the existence of an increasing sequence (nj) of integers such that
a = limfc>,·_>„ Un.nk exists uniformly in к >j. Put gj =/„..
We shall prove a = 0. If this fails, there exists, for all ε > 0, an integer Μ such
that
V(Jfc, /) g ^2, Μ < к < I => J I \gk - gl| - a\dl < г2.
For such a pair (/c, /) we have
4\\gk-gi\-a\>s)<8,
and one can choose four integers f <j2 <j3 <j\ such that the six sets
{Igjr — gja\ — аI й β} have Я-measure ^ 1 — ε. It follows that there exists at least
one point ωβΩ for which the numbers zr = gjr(pji) satisfy the following
inequalities:
\\zr-zs\-a\us, (l^r,*^4).
For α Φ 0 and sufficiently small ε > 0 this yields a contradiction. Thus (/„.) is a
Cauchy sequence, α
If T: Li -* Lx is conservative and ergodic we have seen that the assumption "Ω
is special" implies the existence of an invariant probability measure λ. Put
1° = (/ - 1 ® X)LX and L% = (I- 1 ® X)LO0. If 11 is special and we take θ = 1 in
the proof of theorem 17, then we have ΘΙΓΘ = Ρ' = Ρ with A = Ω. The proof then
shows that /— Pn is an automorphism of L° for some n. (By an automorphism
we mean a bijective linear continuous operator with continuous inverse.) Then /
- pn = (I- p) (J + ρ + ... + ρ»-1) implies that I-P must be an
automorphism of L°. By duality I— Tmust be an automorphism of L?.
We shall now prove:

Supplement: Harris processes 317
Proposition 24. IfI— Τ is an automorphism of L°x and zeC with | ζ | = 1 is no
eigenvalue of T, then any L^bounded sequence (fn) with
lim || (zl — T) (fn —fp)Hi = 0 is a Cauchy sequence.
n,p-> oo
Proof Put φ„,ρ:= \fn -fp\ - I T(fn -fp)\. We have wny.= T\fn -fp\ - \fn -fp\
+ φη,ρ ^ 0 and \w^pdk = \φη,ράλ.
From |z| = 1 we obtain |φπ>ρ| й \(zl — T)(fn—fp)\ and hence
Ь™п,р^ао II (Pn,p\\i = 0. This shows wHfP - φη,ρ = T(fn -fp) - (fn -fp) ^ 0, (in
Lx). Consequently,
lim ||(/-Γ)(ΐΛ-ΛΙ-ίΙΛ-ΛΙ)ΙΙι = θ·
n,p-> oo
As Ι— Γ is an automorphism in L°u this implies |||/„ — fp\ — J \fn — fp\\\i -> 0.
Now lemma 23 shows the existence of a subsequence of (fn) which converges in
Lx. If the sequence (/„) is not Cauchy we can extract two subsequences
converging to different limits h, h'. But then T(h — h') = z(h — h) contradicts the
assumption that ζ is not an eigenvalue of Τ. α
We can now deduce
Proposition 25. If I — Τ is an automorphism ofL\, the only points on the unit circle
which belong to the spectrum of Τ are the eigenvalues.
Proof. Let us first show that (zl — T) Lx is a closed subspace if ζ with | ζ | = 1 is not
an eigenvalue of T. It is enough to show that a sequence (/„) for which {zl — T)fn
converges in Lx to some φ must be bounded.
If 11/J | is not bounded there exists a subsequence with ||/w.|| -»oo. Then
fi—fm/WfrnW satisfies (zl— T)f{ -* 0 in Lx and proposition 23 shows that//
converges in Lx to some h. Clearly, \\h\\ = 1 and h is an eigenvector with
eigenvalue z, in contradiction to our assumption. To complete the proof let us show
(zl— T)LX = Lx. If this fails there exists 0 φ и e L^ orthogonal to (zl— T)LX.
This would imply T*u = Pu = zu. Thus, ζ would be eigenvalue of Ρ and hence of
Т. и
Now everything is ready for the analysis of the structure of T.
Theorem 26. Τ is quasi-compact if I— T is an automorphism of L\.
Proof. Let us first remark that if /— Τ is an automorphism of L°X(X), with λ a
probability measure, then λ is invariant. This follows since the fact that I — Pis
an automorphism of L%(X) implies (/— P) (f— \fdX) e L° for all/e L£, and
hence \PfdX = \fdX. Now, if ζ Φ 1 with |z| = 1 is an eigenvalue of Τ and
Th = zh, 0 φ h e Lx, one has \hdk = J ThdX = z\hdX. Hence \hdX = 0 and

318 Supplement: Harris processes
A e L?. Considering the restriction of Γ to L? and writing (zT — I) = (z — I) I + I
— Τ we see that
(z/- T) (I- ТУ1 = /+ (ζ - 1) (/- ТУ1 on L°.
We can infer that the restriction of (zl — T) to L? is invertible if ζ is sufficiently
close to 1. Thus, 1 is an isolated eigenvalue of Ton the unit circle. As the powers
of eigenvalues on the unit circle are eigenvalues on the unit circle the preceding
argument shows that ζ is a root of unity. All eigenvalues of Γ therefore are of the
form exp (2nni/q), η = 0,1,..., q — 1. On the other hand, if Th = zA, Α φ 0, one
has T\h\^\Th\ = \h\ and therefore | h \ — constant > 0. We can assume | h | = 1.
Passing to the reference measure λ we can also assume 71 = 11 = P1. We can
apply the results of Akcoglu-Brunel (from Section 4.1) to zT9 where
ζ = exp (2ni/q). Clearly, the linear moduls of ζ 74s T. Thus, there exists a
measurable function s of modulus 1 such that, for all φ e Ll9
ζΤφ = ^Γ(^φ).
For φ = H we obtain Ts — zs and Ts = zs. One also has
zTs = sT(s2) = z2s and T(s2) = z2s2.
By recurrence we deduce T(s") = zns". For η = q, T{sq) = sq, which implies
sq = 1 = sq. Hence s is of the form
s= Σ iAnexp(2nin/q)
n = 0
where A0, Al9..., An-1 is a measurable partition of the space. Writing Aq ··= A0
we find
(3) Ts = Σ exp (2nin/q) T\An = zs = Σ exp (2πι(/ι - l)/9) 1 An
9-1
= Σ txp(2nin/q)iAn+l.
n = 0
We shall now show that (3) implies 71 An = 1 An +1? (« = 0,..., q — 1). We multiply
both sides of (3) with \Ah + 1,Q ^k ^q — 1, and integrate with A. Then we find
«-ι
Σ exp(2nin/q) \\Ah+J\AJX = ехр(27пВДА(Лк+1).
и = 0
This yields
ί (*Σ exp (l™"-^) iA„)T*iAk+ldX = Α(Λ+ι)
«=o \ Я J
= i^U+1^ = i{?cos^^)l^|r*U+lu?A.

Supplement: Harris processes 319
As{...}is S 1, it must be = 1 on {T*iAk+l > 0}. Ascos(27r(n - k)/q) < 1 holds
for η φ к, we necessarily have T*lAk+lt^ lAk> (& = 0, ...,#-1). But now
q-l q-1
Ц = г*И = Σ T*iAk+1<Z Σ lAk = 1 shows that the desired equalities must
hold. k = 0 k = 0
It follows from T\An = 1 An +1? (л = 0,..., q - 1), that 7МАо = lAo. Therefore,
Г maps Ll(A0) into Lt(A0). Let 5 denote the restriction of Tq to LX(A0). We
want to show that IAo — S is an automorphism of L°(A0). It is clear that S is
conservative and ergodic and that IAo — S is injective and has a dense image in
«-ι
L? (Л)· It remains to show that (/Αο - S) L°x (A0) is closed. Put R = Σ Tk. Then
fc = 0
R maps L\{A0) into L?((2) and/= lAoi?/holds for/e L?(^0). Now, if (/w) is a
bounded sequence in L?04o) with (IAo — iS)/„ -> 0 in L%(A0)9 then the identity
(/- T)Rfn = (/- F)/„ = (U " S)/»
shows that (I-T)Rfn tends to 0 in L?(0). Hence Н-ВД^О, and fn
= \AoRfn -> 0 in L?(v40). We shall show that Sand hence Tis quasi-compact. We
can consider Г and can assume that 1 is the only eigenvalue. Take Ε = 11 ® λ. On
L? we have T= T— E. The point 1 lies outside the spectrum of Γ considered as
an operator in L? and its spectral radius is < 1. It follows that
lim||(r-£)w||Lo = 0.
η
For/e Lx we can write f = f + к with/'e L? and fceC. Thus
(Г- E)f= (T- E)f = Г/ - £/' = Г/'.
We obtain (T-E)nL1 = (Г- £)nL?. It then follows that
lim||(r-£)n||Ll = lim||(r-£r||Lo = 0.
As (T-E)n = Tn-E, (Ti = H), we arrive at lim || Tn - E\\Ll = 0. In particular,
Τ is quasi-compact, α
Notes
It has been shown in the studies of Fortet [1938] and Yosida [1939] that the validity of the
uniform ergodic theorem for Markov chains is closely related to the property of quasi-
compactness of the transition kernel. Doeblin [1940] and Harris [1956] have given the
appropriate recurrence conditions and Harris showed that his condition (FT) implies the
existence of a σ-finite invariant measure.
Orey [1962] proved a fundamental theorem, generalized by Jamison and Orey [1967]
from discrete to general state spaces: If Τ is an "aperiodic" Harris operator and ν is a
signed measure with ν(Ω) = 0 then \\vPn\\ -> 0.
Ornstein [1969 a] proved theorem 17 in the particular case of random walks on U which
are recurrent in the sense of Harris, and this theorem was generalized by Metivier [1969] a
bit later. The work of Orey and Ornstein motivated the theory of "bounded" sets as
developed by Brunei [1971]. They were later called special sets. In particular, it was shown

320 Supplement: Harris processes
that the existence of special sets implies the existence of a σ-finite invariant measure, and
that, for a conservative ergodic T, I — Τ is an automorphism of L? iff Ω is special. (This
theory was presented in a seminar at Minneapolis in 1969).
Horowitz [1969a], [1969b], [1979] proved the principal theorems of the abstract
theory and noticed the connection with the uniform ergodic theorem.
Then, Neveu [1972] has brilliantly generalized the theory of special sets introducing the
special functions and the taboo-operators and their various applications.
Brunei and Revuz [1974] have taken up these notions in their study of the connections
between quasi-compactness and the Harris condition, and they used them to determine
the Martin boundary of Harris recurrent random walks.
In the meantime, Ornstein and Sucheston [1970] had established the important zero-
two law. A unified treatment was given by Foguel [1979] and Foguel-Ghoussoub [1979].
The present proof of theorem 18 follows Pfort [1984] who used their ideas but corrected
an error in their paper.
The author of this supplement has not tried to give the shortest route to the key
theorems. He has made an effort to acquaint the reader with different techniques. The
exposition has been limited to the abstract point of view dealing with the operator ergodic
theorems which form the main subject of this book.
There is a considerable literature on the subjects connected to this supplement, and a
short guide to it shall suffice here. The books of Orey [1971] and Revuz [1975] give an
exposition of the theory of Harris processes from the probabilistic point of view.
The papers of Chung [1964], Jain [1966], Jain-Jamison [1967], Winkler [1975] give
contributions to Doeblin's theory of Markov chains. Other probabilistic approaches to
the limit theorems have been proposed by Papangelou [1976/77], Athreya-Ney [1978],
[1982], Nummelin [1978], Pitman [1974]. A special case was treated by Moy [1965],
[1967].
The papers by Halmos [1947], Dowker [1951], Feldman [1962], Sidak [1967],
Horowitz [1968 a], Millet-Sucheston [1979] are references concerning existence of σ-
finite invariant measures; see also § 3.4 and § 3.6. Millet-Sucheston considered small sets
analogous to special sets.
Theorem 17 has been a useful tool for the proof of "global" or "mixed" ratio theorems,
in which one considers ratios of the form
£<v1,Pih1}/i<V2,Pih2>.
i = 0 i = 0
(Note that if h = 1A, then <v, Plh} is the probability of a visit in A at time /when the initial
distribution is v.)
For this topic we refer to Krengel [1966], Isaac [1967], Levitan [1970], [1971], Lin
[1970a], [1972b], [1976], Foguel-Lin [1972] and Metivier [1972].
The zero-two law is closely connected to the triviality of the remote σ-algebra,
established for Harris processes by Jamison-Orey [1967]. Related references are the papers of
Winkler [1973], Derriennic [1976], Foguel [1976], Cohn [1979a], Lin [1982a], Revuz
[1983] and Greiner-Nagel [1982].

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AM St
Bull AMS
Can. JM
CRAS
Ergebn. Μ
Isr. JM
J
JMS Jap.
Ann. Inst. Henri Poincare
Ann. Math. Stat.
Bull. Amer. Math. Soc.
Canadian Journal Math.
Comptes Rendus Acad. Sci. Paris
. Ergebnisse d. Math. u. Grenzgeb.
Israel Journal Math.
Journal
Journal Math. Soc. Japan
Μ
MZ
Рас. JM
PAMS
SpLNM
TAMS
Ton. MJ
ZW
Math.
Math. Zeitschr.
Pacific Journal Math.
Proc. Amer. Math. Soc.
Springer Lecture Notes Math.
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Notation
Notation used only in a single section need not appear in this list
General Nc
3
V
:=
=>
<=>
iff
D
и
ζ
Ν
Q
С
V
αν b
a a b
a+
a~
Re()
Ac
cl
CO
CO
lin
card (A)
1
F1
[w, v\_
</,*>
AAB
I
t
и
x+
A® B
ker
dom
C(Q)
station
there exists
for all
equal by definition
logical implication
logical equivalence
if and only if
end of proof
reals
integers
{1,2,...}
rationals
complex numbers
{0,1,2,...}', 195
Max (a, b)
Min (a, b)
flvO
(-a) vO
real part
complement of A
closure, 4, 71
convex hull, 71
cl со, 71
linear span, 71
number of elements of A
orthogonality, 4
orthogonal complement
of F
interval, 195
scalar product, 3, 71
symmetric difference, 3
identity, 3
constant function 1, 3
indicator function of A,
3
positive cone of X, 3
direct sum, 71
kernel, 80, 83
domain, 83
continuous functions, 2
Cb(Q)
w-lim
w*-lim
wo-lim,
w*o-lim,
so-lim
l.c.t.v.s.
IIЛ1
X*
τ*
EJ
&{J)
el(=e(i))
σ(Τ)
r(T)
RX,R(KT)
bounded continuous
functions, 111, 223
weak limit, 71
weak*-limit, 71
limits in the weak,
weak* and strong
operator topologies, 80
75
norm of operator T, 3
adjoint (dual) space, 71
adjoint operator, 3, 71
is-valued functions on /,
22
bounded linear
operators in X, 3
z'-th unit vector in Ud,
206, 289
spectrum, 90
spectral radius, 90
resolvents, 83, 90
Measure theory
£>ρ = £>ρ(Ω,^,μ)2
11/11,
Lp = Lp(Q,s/,
LP{B)
**
%
ν <μ
ν ~ μ
v = gμ = g· μ
Li
L\ogmL
εω
E(f\&)
= Εμ(/\&)
= E*f
2
μ) 2
{feLp:{f*0}aB},
111, 137
Lp with counting
measure on Ζ +
167
3
3
dv/άμ = g, 127
{gp.geLJ, 114
51,54
measure with mass 1 in
ω, 177
conditional expectation,
5

348 Notation
bs/'
Μ = Μ {si)
Jit
μχν,Πμ]
s/ ® #, s#3
Basic ergodic
— 1 / л\
μ°τ ι(Α)
S„f= 5„(Γ)/
AJ= An(T)f
An, Sn
msj
MJ
Sa,bf
As,t
π (κ)
^Ο,οο/
кш
M^Mf
J
τΑ
νΑ>νΑ(ω)
^ret? ^inf
С
<e
D
С
D
bounded ^/-measurable
functions, 113
signed measures on s/,
113
probability measures on
si
product measure, 10, 23
product-a-algebra, 10,
23
theoretic notation
μ{τ-γΑ\ 1
"Σ Tkf, 4
k = 0
n~lSnf,4
in multiparameter case,
195
Max (Si/,..., SJ), 7
Maxk<^k/, 7
b
J !ζ/Λ, 58, 216
a
(t-sy1SStt, 83,
multiparameter case, 216
Ui · u2 '... щ for и
= (ul9u2, ...,wd), 195
SUp {v40,a/: 0 < a < OO},
59
58
63
invariant σ-algebra, 5,
203
induced transformation,
20
return time, 19
16
conservative part, 16,
17, 116, 117
absorbing sets in C, 126
dissipative part, 17, 116
maximal support of
finite invariant measure,
141
Q\C, 141
Mean ergodic theory
F=F(T),F(y) fixed space, 73, 77
^*> ^*(T)> ^c(^) fixed space of adjoint
operator or semigroup,
73, 77
N,N*
BE
£//
BE
^uds
3E
D*(M)
А-(АО
D (M)
Z>-lim
ll-H-co
Processes
*i(fl>)
s4{m,rt)
s/^sZ-n
PA
Ρ(ω, A)
Fi,k> pi
F,
Contractions ι
h,h
II rili..
^£j ix
П,*н
Ψε,Ψε
нЕ
Т,|Г|
P*g
Η
uh
uA
Q)? ^Л)з С 3 ^
6
1Я
73, 77
mean ergodic subspace,
73
flight vectors, 94, 105
reversible vectors, 105
subspace with uni-
modular discrete
spectrum, 94, 106
almost periodic vectors,
110
upper density, 95
lower density, 95
density, 95
convergence in density,
95
87
i-th coordinate of ω, 22
^+,23
24
remote σ-algebras, 27
conditional probability,
28
substochastic kernel,
113
sub-o'r superadditive
processes, 35, 201
X(I)-lFj, 202
in Lt
multiplication by h, by
1£, 124, 301
51
124
122
125
126
linear modulus, 159
120
non zero measurable
functions h: Ω -► [0,1].
taboo potential
operator, 301
303
230
special
functions, 305
{/el^J/rf^O}.

Notation 349
Other special notation
A\A2
Μ
A*
P.(4, ω)
л*<*
Алк
β
®ь
Pd, P°
5
m χ ra-matrices, 42
transposed matrix, 42
42
43
44
201
208
243
251
я
91
Ksa
Я*
Щ
%
S<P
S
mm, ι (ξ)
Η(ξ),Η(ξ\!8)
strictly increasing
sequences in Z+, 251
267
268
268
269
269
269
state space, 274
281
281

Index
(In general, names are listed only for detailed quotations.)
Aaronson, 15, 256
Abel bounded, 83
Abelian averages, 257
Abel transformation, 257
absorbing projection, 74
absorbing set, 5, 118
additive, 35, 147, 202, 223, 229
adjoint, 71
admissible, 129, 164
affine, 81, 85
Akcoglu, 36, 96, 146, 163, 186ff, 192,
203rT, 220, 231, 236, 238, 249, 250,
253, 256.
Alaoglu, 80
almost convergence, 291
almost periodic, 76, 103, 110
almost separably valued, 60
almost subadditive, 288
almost surely, 24
almost uniform convergence, 273
almost uniform ergodic theorem, 274
oc^-contraction, 270
amenable, 223
Anshelevich, 279
Artstein, 297
asymptotically invariant, 134
asymptotically isometric, 292
asymptotically mean stationary, 32, 265
asymptotic center, 295
asymptotic dominance, 33
automorphism, 1
*-automorphism, 268
Baillon, 288, 290, 295
balayage operator, 303
Banach algebra, 40, 267
Banach limit, 2, 135
Banach principle, 64
barycenter, 274
Baxter, 62
Bellow, 226, 263 (see: A. Ionescu Tulcea)
Bernoulli shift, 23
Besicovich sequence, 264
bilateral, 22
bilateral extension, 29
Birkhoff, Garrett, 80
Birkhoff, George, 9, 13, 26, 65
Bishop, 14, 166
Blackwell, 28
Blum, 225, 253
Bochner, integral, 167
Bogoliouboff, 88
Boolean algebra, 16
Borel, 29, 33
bounded operator, 3
bounded set of operators, 75
Breiman, 182, 284
Brezis, 290
Browder, 290
Bruck, 292
Brunei, 82, 92, 125, 145, 163, 21 Iff, 217,
260, 310, 316
Brunei function, 125
Burkholder, 61, 68, 191
Butzer, 84
C*-algebra, 267
canonical representation, 27
canonical weights, 251
Cesaro bounded, 72
Chacon, 119rT, 134, 151, 161, 164, 169,
231, 238
characteristic exponent, 43
Choquet, 184
Chung, 281, 287
cluster process, 202
cocycle, 50
compact operator, 88
complete, 77
completely mixing, 254

352 Index
composition, 3
conditional entropy, 281
conditional expectation, 5, 167
conditional information, 281
conditional probability, 30
conservative, 16, 17, 116, 117
consistent, 29
continuity principle, 68
continuous in measure, 63
continuous spectrum, 97
contraction, 3, 270
convergence in density, 95
convex hull, 71
convex operator, 296
convolution, 119
Conze, 259, 263, 274
Cotlar, 62
countably order complete, 171
Dang-Ngoc, 274
Davis, 61
Day, 208, 225
de la Torre, 62, 186
Deleeuw, 105, 106, 111, 182
del Junco, 249
density, (upper, lower), 95
Derndinger, 225
Derriennic, 61, 84, 172, 220
dilation, 192
Dinges, 157
directed set, 63
direct product of endomorphisms, 98
direct sum, 71
disappearing part, 172
dissipative, 17, 116, 117
distal, 86
distribution, 23
Djafari-Rouhani, 296
Doeblin, 123, 320
dominated ergodic theorem, 52, 59, 189,
190, 224, 258
Dotson, 85
Dowker, 32, 145, 175
dual operator, 3, 71, 131
Dubins, 227
Dunford, 53, 60, 65, 131, 161, 196, 217,
244
dyadic rationals, 58
Eberlein, 73, 76, 90
eigenoperator, 103
eigenvector for a semigroup, 106
Eisenberg, 225
Emerson, 227
Emilion, 84
empirical distribution function, 297
endomorphism, 1
entropy, 281
equicontinuous, 12, 75
equilibrium potential, 125
equivalent measure, 3
ergodic, 6, 26, 126, 203
ergodic decomposition, 155, 166, 185
ergodic hypothesis, 6
ergodic net, 75, 87, 251
ergodic square function, 62
ergodic theorem for seqences of
functions, 66
essential limit, 242
essential supremum, 242
essential total variation, 237
exact dominant, 147, 150
exact endomorphism, see e. g. Cornfeld
et al (1982)
exhaustion argument, 17
exhaustive weakly wandering set, 144
exterior power, 43, 44
face, 276
faithful, 269
Falkowitz, 82
Fava, 196 if
Fejer, 76
Feldman, 18, 134, 145
Feyel, 262 if
Fichtner, 299
filling operator, 120
filling scheme, 120
filtered flow, 34
filtration, 43
finitely additive measures, 22
finitely valued, 60, 167
fixed point, 77
fixed point theorem, 81, 86, 296
flight vector, 94, 105
flow, 10
flow under function, 34
Follmer, 282
Foguel, 304, 306, 309, 314
Foias, 184, 193
Fong, 157, 173, 174
Fourier transform, 96
Frechet differentiable norm, 296
Freedman, 28

Index 353
Freudenberg, 299
Friedman, 259
Fritz, 208, 209, 282
Furstenberg, 40, 99
Garsia, 68
Gaussian process, 30
generalized summing sequence, 225
generator (for a m.p.trsf.), 35
generator (of a semigroup), 83
generic, 99
Ghoussoub, 171, 304, 309
Giroux, Gaston, 134
Glicksberg, 105, 106, 111, 182
global ratio theorem, 320
Gol'dshtejn, 280
Gray, 32
Greenleaf, 223, 227
Grillenberger, 211, 263
Grimmett, 202
Hajian, 144, 221
Halasz, 14, 56
Halmos, 18
Hanson, 252, 253
Hardy, 50, 53, 262
harmonic, 116, 172 if
Harris, 119, 301 ff
Harris operator, 304
Harris recurrent, 301
Hausdorff metric, 297
Helmberg, 21, 175
Hubert transform, 62
Hille, 83
homomorphism, 1
*-homomorphism, 268
Hopf, 8, 17, 58, 65, 128, 254
Hopf decomposition, 17, 116
Horowitz, 306, 320
Hurewicz, 123
ideal, 94, 104
identically distributed, 24
i.i.d. case, 61
incompressible, 16
independent, 23, 48
indicator function, 3
induced contraction in Lp, 179
induced operator on subset, 134, 303
induced transformation, 20
information, 281
initial distribution, 31
initially conservative, 230
initially dissipative, 230
initially invariant, 239
integral kernel, 304
integrally independent, 12
invariant function, 5, 131
invariant initial distribution, 31
invariant mean, 108, 111, 223
invariant measure, 1, 131
invariant set, 5, 26
invariant state, 268
involution, 267
Ionescu Tulcea, A. ( = Bellow), 171, 174,
175, 186, 284
Ionescu Tulcea, C.T., 93, 171
Irmisch, 263
irreducible, 94, 179
isometry, 3
isomorphic transformations, 21
Ito, Y., 145, 221
Jacobs, 22, 105, 106, 111, 155, 182, 264
Jajte, 171, 280
Jamison, 180
Jessen, 227, 250
jointly continuous, 103
Jones, L.K., 109ff, 144, 252, 253, 255
Jones, R.L., 61, 62
Kac, 19, 55, 134
Kakutani, 8, 20, 73, 81, 82, 90, 91, 144,
296
Kan, 186
Kantorovic, 161
Karlin, 93
Katznelson, 13, 99, 282
K-automorphism, see Petersen (1983)
Keane, 260
kernel, 80, 104
Kesten, 40, 48
Khintchine, 13, 22
Kieffer, 32, 265
Kingman, 37, 41
Kingman decomposition, 147
Kipnis, 246
Koliha, 79
Kolmogorov, 24, 28, 67
Koopman, 96 \
Kovacz, 269, 272
Krengel, 22, 36, 62, 84, 86, 102, 143,
144, 161, 203ff, 220, 226, 231, 236,
250, 254, 259

354 Index
Kryloff, 88
Kubokawa, 246
Kummerer, 272, 274
Kuftinec, 253
Lamperti operator, 193
Lance, 269, 274
Lebesgue, 244
Lebesgue spectrum, 259
Lin, 65, 85, 87, 92, 93, 108ff, 172, 175,
239, 240, 252, 256
linear modulus, 159
linear span, 71
linearly bounded, 236
Littlewood, 50, 53, 262
Lj-L^-contraction, 51
local ergodic theorem, 11
locally bounded, 58, 243
locally convex, 75
locally finite, 63
logic, 16
Lorch, 73, 74
Lorentz, 291
Lotz, 93
majorizable, 161
Maker, 66
Marcinkiewicz, 54, 250
Marcus, 56, 57, 61
marginal distribution, 23
Marinescu, 93
Markov, 81
Markov chain, 30
Markov operator, 115, 177
Markov property, 30
Markov shift, 31, 119
Marstrand, 227
martingale, 171, 298
Masani, 84
maximal ergodic inequality, 8, 51, 67,
205
maximal ergodic theorem (lemma), 8,
169, 223, 262, 275
maximal function, 50
maximal operator, 52, 197
maximal type relation, 52, 54
Mazur, 72
Mc Grath, 196, 245, 250
Mc Millan, 282
mean ergodic, 73
mean ergodic theorem, 4, 72, 174, 203,
220, 222
mean ergodic theorem for information,
283
measurable flow, 10
measurable semigroup, 223
measurable space, 1
measure preserving, 1
measure space, 1
Menchoff, 191
Metivier, 310
metrically transitive, 7
Meyer, 134, 156
mild mixing, 103
minimal, 180
mixing, 24, 102, 254
modification, 156
mod μ, 2
Moretz, 174, 175
moving average, 26
Moy, 54, 119
multiple recurrence, 99
multiplicative ergodic theorem, 42, 49
μ-continuous, 3
Nagel, 79, 80, 225, 272
name, 285
von Neumann, 4, 96
von Neumann algebra, 268
Neumann, Norbert, 212
Neveu, 127, 145, 154, 281, 313, 320
Nguyen, 209, 210
Niiro, 93
nonexpansive, 288
non homogeneous chains, 298
nonlinear averages, 299
nonlinear operator, 288 if
non negative definite, 94
nonsingular, 3
non trivial kernel, 304
norm, 3
normal, 268
null part, 141
null preserving, 3, 114
O'Brien, 14
odd, 294
Opial's condition, 296
order continuous norm, 171
Orey, 319
Ornstein, 54, 119, if, 153, 157, 242, 259,
284, 310, 315
oscillation, 41

Index 355
Oseledec, 42
Oseledec space, 50
Palm, 225
parameter space, 22
partial application of T, 123
partial mixing, 103
partition, 21, 281
percolation, 39
Petersen, 56, 57, 61
Pettis, 60
Phillips, 83
Pitman, 227
Pledger, 252
Poincare, 17
pointwise ergodic theorem for
information, 284
polar decomposition, 43,
pole, 90
Polish space, 29
positive element of C*-algebra, 268,
positive operator, 3
positive part, 141
positive semidefinite, 43
positive type, 144
power bounded, 71
pre-adjoint, 270
predual, 268
process, 229
product measure, 23
product-a-algebra, 23
projection, 74, 269
proper pseudo resolvent, 262
pseudo resolvent, 83
Q-lim, 230
quasi-compact, 88, 313
quasi-regular, 179
quasi-Stonean, 185
Rahe, 265
random ergodic theorem, 261
random sets, 297
range of random walk, 39
ratio ergodic theorem, 14, 129, 150,
rational rotations, 227
recurrence theorem, 16, 19
recurrence times, 28
recurrent, 16, 117
recurrent in the sense of Harris, 301
recurrent state, 31
recurrent weights, 263
reference measure, 114, 128
refinement, 281
regular, 203, 209
regular operator, 161
Reich, 295, 296
remaining part, 172
remaining in a sector, 203
remotely trivial, 102
remote σ-algebra, 27, 48, 119
representative, 230
residue, 90
resolvent, 83, 90
resolvent equation, 83, 301
resolvent set, 90
return time, 19
reversible, 105
Revuz, 92, 316
Reynolds operator, 298
Riesz, 73
rigid, 103
Robbins, 119
Rottger, 62
Rost, 155
Rota, 298
Rudin, 227
Ryll-Nardzewski, 86
Sato, 79, 175, 231, 238, 240
Sawashima, 93
Sawyer, 69
Schaefer, 93
Scheller, 134
Schurger, 297
Schwartz, 53, 60, 65, 131, 161, 217, 244
sector, 203
self-adjoint, 268
semiflow, 10
seminorm, 75
semitopological semigroup, 103
separable, 41
separately continuous, 103
^-ergodic net, 75, 251
Shannon, 282
shift, 23
Siegmund, 62
σ-algebra, l
σ-topology, 269
Simons, 119
Sinai, 7, 279
Sine, 74, 82, 85, 175, 296
skew product, 261
skyscraper, 21

356 Index
Smythe, 196,203,211
spatial constant, 202
special function, 305
special set, 305
spectral measure, 95
spectral radius, 90
spectrum, 43, 90
speed of convergence, 14, 84
stable, 34
stacking construction, 153
Stadje, 238
Starr, 192
state, 268
state space, 23, 274
stationary, 24, 25
stationary in the wide sense, 32
stationary mean, 32
stationary transition probability, 30
Steele, 171
Stein, 67, 190
stochastically convex, 68
stochastic convergence, 143
stochastic ergodic theorem, 143
stochastic kernel, 113
stochastic matrix, 30
stochastic partial order, 49
stochastic process, 23
strong law of large numbers, 24, 33
strongly conservative, 141
strongly continuous process, 229
strongly continuous semigroup, 58, 216,
229, 294
strongly measurable, 60, 167
strongly subadditive, 211
strong type, 53
subadditive, 35, 39, 40, 75, 146, 171, 202.
223, 249, 280, 297
subinvariant, 131
sublinear, 53
sub-Markovian, 115
sub-orbit, 265
substochastic kernel, 113
Sucheston, 66, 96, 146, 157, 173, 175
192, 196, 220, 253, 254, 256, 315, 316
Sucheston decomposition, 172, 193
superabelian process, 263
superadditive, 35, 146, 201, 223, 249
2-superadditive, 202
superharmonic, 116
super-orbit, 265
superstationary, 49
support, 269
symmetric difference, 3
Sz. Nagy, 193
Szucs, 269, 272
taboo potential, 301
tail-a-algebra, 27
Takahashi, 295
Tauberian theorem, 262
Teicher, 62
Tempel'man, 205, 209, 224, 226, 248
Terrell, 244, 245
Thouvenot, 282
time constant, 35
topological semigroup, 103
topological vector space, 75
topologies:
norm, 71
strong, 71, 269
strong operator t., 80
s-topology, 269
s*-topology, 269
σ-topology, 269
uniform t., 269
uniform operator t., 86
weak t, 71
weak operator t., 80
w*-topology, 71
w*-operator t., 80
total variation, 237
transient, 31, 117
trivial σ-algebra, 27
Tucker, 297
uniform distribution mod 1, 13, 16
uniform ergodic theorem, 87, 88, 91
uniformly ergodic, 86, 87
uniformly integrable, 132
uniform sequence, 260
unilateral, 22
unimodular discrete spectrum, 94, 106
uniquely ergodic, 178
unital, 268
unitary subspace, 110
universally bad sequence, 259
universally good sequence, 260
unrestricted convergence, 195
upcrossing inequalities, 14, 48, 49, 166
Vitale, 297
Wacker, 49, 62, 249
wandering, 16

Index 357
weakly almost periodic, 103
weakly independent, 102
weakly measurable, 60, 221
weakly mixing, 97, 254
weakly stable, 103
weakly wandering, 138, 143, 218
weak type, 53
weighted conservative part, 146
Weiss, 256, 282, 284
Westphal, 84
Weyl, 13
Wiener, 8, 11, 14,96,203
Wintner, 14
Yeadon, 280
Yosida, 8, 73, 90, 91
zero-one law, 28
zero-two law, 309 if
Zessin, 209, 210
Zygmund, 51, 196ff, 245, 250