Stopping Times and Directed Processes (Encyclopedia of Mathematics and its Applications)

This book applies the technique of stopping times to convergence problems in probability and analysis. There are applications in sequential analysis. Convergence of stochastic processes indexed by directed sets is studied and solutions are given to problems left open in Krickeberg's theory of martingales and submartingales. The rewording of Vitali covering conditions in terms of stopping times establishes connections with the theory of stochastic processes and derivation. A study of martingales yields laws of large numbers for martingale differences, with application to "star-mixing" processes. Convergence of processes taking values in Banach spaces is related to geometric properties of these spaces. There is a self-contained section on operator ergodic theorems: the superadditive, Chacon-Ornstein, and Chacon theorems. A unified treatment of martingale and ergodic theory is in part based on a "three-function inequality."

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

EDITED BY G.-C. ROTA Volume 47

Stopping Times and Directed Processes

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS 4 W. Miller, Jr. Symmetry and Separation of Variables D. Ruelle Thermodynamic Formalism: The Mathematical Structures of Classical Equilibrium Statistical Mechanics 6 H. Minc Permanents 7 F. S. Roberts Measurement Theory with Applications to Decisionmaking, Utility, and the Social Services II W. B. Jones and W. J. Thron Continued Fractions: Analytic Theory and Applications 12 N. F. G. Martin and J. W. England Mathematical Theory of Entropy 15 E. C. Beltrametti and G. Cassinelli The Logic of Quantum Mechanics 17 M. Lothaire Combinatorics on Words 18 H. O. Fattorini The Cauchy Problem 19 G. G. Lorentz, K. Jetter, and S. D. Riemenschneider Birkhoff Interpolation 21 W. T. Tutte Graph Theory 22 J. R. Bastida Field Extensions and Galois Theory 23 J. R. Cannon The One-Dimensional Heat Equation 24 S. Wagon The Banach-Tarski Paradox 25 A. Salomaa Computation and Automata 26 N. White (ed.) Theory of Matroids 27 N. Bingham, C. Goldie, and J. L. Teugels Regular Variation 28 P. Petrushev and P. Popov Rational Approximation of Real Variables 29 N. White (ed.) Combinatorial Geometries 30 M. Pohst and H. Zassenhaus Algorithmic Algebraic Number Theory 31 J. Aczel and J. D. Hombres Functional Equations Containing Several Variables 32 M. Kuczma, B. Chozewski, and R. Ger Iterative Functional Equations 33 R. V. Ambartzumian Factorization Calculus and Geometric Probability 34 G. Gripenberg, S.-O. Londen, and O. Staffans Volterra Integral and Functional Equations 35 G. Gasper and M. Rahman Basic Hypergeometric Series 36 E. Torgersen Comparison of Statistical Experiments 37 A. Neumaier Interval Methods for Systems Equations 38 N. Korneichuk Exact Constants in Approximation Theory 39 R. Brualdi and H. Ryser Combinational Matrix Theory 40 N. White (ed.) Matroid Applications 41 S. Sakai Operator Algebras in Dynamical Systems 42 W. Hodges Model Theory 43 H. Stahl and V. Totik General Orthogonal Polynomials 44 R. Schneider Convex Bodies 45 G. Da Prato and J. Zabcyck Stochastic Equations in Infinite Dimensions 46 A. Bjbmer et al. Oriented Matroids 47 G. A. Edgar and L. Sucheston Stopping Times and Directed Processes 48 C. Sims Computation with Finitely Presented Groups 5

ENCYCLOPEDIA OF MATHEMATICS AND ITS APPLICATIONS

Stopping Times and Directed Processes G. A. EDGAR AND LOUIS SUCHESTON Department of Mathematics The Ohio State University

CAMBRIDGE UNIVERSITY PRESS

CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, Sao Paulo

Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK

Published in the United States of America by Cambridge University Press, New York www.cambridge.org Information on this title: www.cambridge.org/9780521350235

© Cambridge University Press 1992

This publication is in copyright. Subject to statutory exception and to the provisions of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. First published 1992

A catalogue record for this publication is available from the British Library

Library of Congress Cataloguing in Publication data Edgar, Gerald A., 1949Stopping times and directed processes / G. A. Edgar and Louis Sucheston. p. cm. - (Encyclopedia of mathematics and its applications; v. 47)

Includes bibliographical references and index. ISBN 0-521-35023-9 1. Convergence. 2. Probabilities. 3. Martingales. 1. Sucheston, Louis. II. Title. III. Series. QA273.43.E34 1992 519.2'6 - dc20 91-44388 CIP ISBN 978-0-521-35023-5 hardback

Transferred to digital printing 2008

Contents

Keywords are shown to provide a fuller description of what is in a section.

Preface

page ix

1. Stopping times

1

1.1. Definitions

2

Moore-Smith convergence directed sets . . dual directed set . . net sequential sufficiency theorem stochastic bases stopping time waiting lemma . optional stopping maximal inequality.

1.2. The amatt convergence theorem smart theorem

.

lattice property semiamart . . smart convergence theorem.

9 cluster point approximation

1.3. Directed processes and the Radon-Nikodym theorem

13

processes indexed by directed sets smarts and ordered smarts set functions . . . Radon-Nikodym theorem. stochastic convergence

1.4. Conditional expectations

19

basic properties of conditional expectations . . . localization theorem martingales . . . submartingales and supermartingales quasimartingales the difference property.

2. Infinite measure and Orlicz spaces 2.1. Orlicz spaces

33 34

Young partition . . . Young's Orlicz function . . Orlicz derivative inequality . . Orlicz modular Orlicz space . . Orlicz heart Luxemburg norm (A2) condition.

2.2. More on Orlicz spaces

51

spaces Lp . . comparison of Orlicz spaces . . largest and smallest Orlicz Orlicz norm spaces . . conjugate Orlicz function . . duality . heart of a sum of Orlicz spaces Fava's spaces Rk.

2.3. Uniform integrability and conditional expectation uniform absolute continuity criterion of de uniform integrability La Vallee Poussin . . . conditional expectation in finite measure space Jensen's inequality.

68

Contents

vi

3. Inequalities

82

3.1. The three-function inequality

82 for Lp and Llogk L

three-function inequality

special cases

two-term reverse inequality

. . maximal inequality for stopped processes.

3.2. Sharp maximal inequality for martingale transforms martingale transform

maximal function

99

Burkholder's inequality.

3.3. Prophet compared to gambler gambler and prophet

103

stopped processes

value of a process

transformed processes.

4. Directed index set

113

4.1. Essential and stochastic convergence . essential and stochastic limits essential supremum Snell envelope. processes indexed by directed set

114 demiconvergence

4.2. The covering condition (V)

127

equivalent formulations .. essential convergence of smarts convergence of martingales.

essential

4.3. Lip-bounded martingales

144

multivalued stopping times covering condition (V,) . . essential condition (FV,). convergence of Lw-bounded martingales

4.4. Ll-bounded martingales condition (C)

..

159

essential convergence of Ll-bounded martingales.

5. Banach-valued random variables

171

5.1. Vector measures and integrals

172

weak and weak-star topologies . . . Hahn-Banach theorems . . . closed random variables graph theorem . Bochner measurability Bochner integral . . . scalarly integrable . . Pettis measurability theorem vector measures . . . variation and semivariation . . . Radon-Nikodym property ... conditional expectation ... Jensen's inequality.

5.2. Martingales and amarts

183

martingale . . . optional sampling . . . uniform smart . . . smart condition Pettis norm difference property . . . Riesz decomposition (B) . . . maximal inequality . . . smart Riesz decomposition.

5.3. The Radon-Nikodym property

198

. Radon-Nikodym set . . . weak a.s. convergence ... weak sequential smart ... strong a.s. convergence martingale (and uniform smart) convergence . . . Pettis norm convergence scalar and Pettis norm convergence T convergence

..

... converse theorems.

vii

Contents

5.4. Geometric properties

218

Ryll-Nardzewski extreme points . . . Choquet-Edgar theorem geometric characterization of fixed-point theorem . . dentable strongly exposed points. Radon-Nikodym property

5.5. Operator ideals

232

absolutely summing operator ... Pietsch factorization ... DvoretzkyRogers lemma . . . amart conditions for absolutely summing operators

Radon-Nikodym operator smart conditions for finite-dimensional spaces amart conditions Lewis-Stegall factorization . . . Asplund operator for separable dual.

6. Martingales

254

6.1. Maximal inequalities for supermartingales

254

maximal inequality for positive supermartingales . . Brunk-Chung-Chow law of large numbers . . application to star mixing.

6.2. Decompositions of submartingales

267

Doob decomposition . . Krickeberg decomposition convergence of martingale transforms.

...

quadratic variation

6.3. The norm of the square function of a martingale square function

maximal function

273

Burkholder inequalities.

6.4. Lifting

280

liftings and densities . . . existence of liftings rem . . . measurability of the upper envelope a stochastic process.

.

Maharam's lifting theoseparable modification of

7. Derivation

291

7.1. Derivation in R

291

difference-quotient martingale

Vitali covering theorem

classical

derivation theorems.

7.2. Derivation in Rd

300

substantial sets .. disks and cubes

intervals.

7.3. Abstract derivation derivation bases density property

.

..

308

strong Vitali property . . . weak Vitali property property (V4,).

7.4. D-bases D-bases . properties (A) and (C) Halo theorems . homothety weak halo evanescence property filling theorem ... weak halo Busemann-Feller bases . . property (WH) . . . properties (WH,y) and (FHq,).

327

Contents

viii

8. Pointwise ergodic theorems

344

8.1. Preliminaries

345

truncated limits .. weak truncated limits .. charges and supercharges

decomposition of set functions.

8.2. Weak maximal inequalities Hopf's maximal ergodic theorem

350 mean bounded operators.

8.3. Hopf's decomposition conservative part

355

... dissipative part

... decomposition theorem.

8.4. The a-algebra of absorbing sets absorbing set

356

... equivalent formulations

... induced operator.

8.5. The Chacon-Ornstein theorem (conservative case) ratio limits

361

identification of the limit.

...

8.6. Superadditive processes

365 ... dominant

superadditive and extended superadditive processes . superadditive ratio theorem exact dominant Dunford-Schwartz theorem . . . Chacon theorem.

..

.

ergodic theorems

9. Multiparameter processes

382

9.1. A multiparameter convergence principle positive operator on Banach lattice

383

convergence principle

Banach-

valued processes.

9.2. Multiparameter Cesaro averages of operators Orlicz spaces Akcoglu . . .

..

multiparameter Dunford-Schwartz

390 .

..

multiparameter

heart of LP + L..

9.3. Multiparameter ratio ergodic theorems sub-Markovian operators

...

394

superadditive ratio limit theorem.

9.4. Multiparameter martingales block martingales . . . block submartingales two-parameter strong law of large numbers

396 .

.. ..

.

reversed martingales alternating procedure.

References

407

Index of names

418

Index of terms

421

Preface

The main themes of this book are: stochastic, almost sure, and essential convergence; stopping times; martingales and amarts; processes indexed by directed sets, multiparameter processes, and Banach-valued processes.

We begin in Chapter 1 with the notion of the stopping time, central to the book. That this notion is important in continuous parameter martingale theory and sequential analysis (briefly touched on in Chapter 3) is well known. This book differs from others in that many of the discrete parameter results are proved via processes (amarts) defined in terms of stopping times-in fact only simple stopping times. The Radon-Nikodym Theorem receives an amart proof. That this theorem follows from the martingale theorem is well known, but here martingales come later, so that the Radon-Nikodym theorem is available to define the conditional expectation and the martingale. In Chapter 4, the rewording of the Vitali covering conditions in terms of stopping times clarifies connections with the theory of stochastic processes. The main result, the Amart Convergence Theorem, is proved by elementary arguments. This-together with a general Sequential Sufficiency Theorem (1.1.3), showing how in metric spaces the convergence of increasing sequences implies that of nets-is used to obtain stochastic convergence of Ll-bounded ordered amarts on directed sets. This in turn implies stochastic convergence of quasimartingales-even on directed sets. Quasimartingales include the Ll-bounded submartingales studied by Krickeberg on directed sets. We believe the proofs by this method are shorter than those existing

in the literature. Chapter 2 is a self-contained presentation of Orlicz spaces. The emphasis is on the "heart" of an Orlicz space-the closure of the integrable. simple functions. The largest Orlicz space is Ll + Lam, and its heart Ro offers the right setting for much of the ergodic and martingale theory in infinite measure spaces. The union of the Lp-spaces, p > 1, used for this purpose in some books (e.g., Dellacherie & Meyer [1982], p. 34), is properly contained

in R0. In multiparameter theory, the space Rk, the heart of the Orlicz space L logk L, plays a major role, as shown in Chapter 9. Except for Fava's multiparameter version of the Dunford-Schwartz theorem in Krengel [1985], this is the first book to treat the theory of spaces Rk.

Chapter 3 deals with inequalities. We claim to offer in this book a partially unified treatment of martingale and ergodic theory. The oneparameter element of the unification is a three-function maximal inequality. Assuming a weak inequality, the passage to the strong maximal inequality

relates appropriate Orlicz spaces and their hearts, and provides a unified

x

Preface

approach that can be used also in harmonic analysis. We do not offer a unified treatment of weak inequalities, usually the most difficult part of a convergence proof. The discussion of a sharp inequality for martingale transforms due to Burkholder [1986] provides an illuminating probabilistic insight. "Prophet inequalities" have an interesting probabilistic interpretation, and seem to fit in the present context, since both stopping times and transforms of processes are considered. Chapter 4 deals with stochastic processes indexed by directed sets. Convergence of martingales and submartingales was studied by Krickeberg in the fifties and early sixties. An exposition is given in the memoir of Kricke-

berg & Pauc [1963] and in the book of Hayes & Pauc [1970]. In that convergence theory, three basic problems remained open: Is the Vitali condition (V) = (V,,,,) necessary for convergence of Ll-bounded martingales? Is the Vitali condition (V°) = (V') necessary for convergence of Ll-bounded submartingales? Is the Vitali condition (Vp) necessary for convergence of

Lq-bounded martingales? (Here 1 < p < oo and 1/p + 1/q = 1.) In all three cases, the sufficiency was established by Krickeberg. All three ques-

tions were answered around 1980: the first two in the negative and the third in the positive. This is the first book that contains an account of this progress, and of subsequent developments. Classes of processes were intro-

duced for whose convergence the conditions (V) and (V°) are necessary and sufficient-amarts and ordered amarts. Amarts may be applied in situations in derivation theory where martingales and submartingales do not suffice. Finally, a covering condition (C) is exhibited, both sufficient and (in the presence of a countable cofinal subset) necessary for convergence of Ll-bounded martingales. The analogous progress in the classical derivation theory is presented in Chapter 7. Here the necessity of the analog of the condition (VP)-and an

Orlicz space generalization-was earlier obtained by C. A. Hayes. Chapter 7 also contains a self-contained presentation of the basics of abstract derivation theory. Similarities between derivation theory and martingale theory on directed sets are easily seen. Chapter 5 deals with Banach spaces. Knowledge of the elements of the theory of Banach spaces is required only for this chapter and parts of Chapter 8. The approach to convergence theorems is geometrical in that we consider convergence theorems that exactly match geometric properties of a Banach space. The best known of these theorems is the characterization of the Radon-Nikodym property due to A. & C. Ionescu Tulcea and S. D. Chatterji. Other more recent geometric characterizations of convergence of martingales are also discussed (5.3.30 and 5.3.34). There are also geometric properties characterized in terms of amarts, such as the Radon-Nikodym property for the dual of the Banach space and reflexivity. Among applications of convergence of martingales we list the Choquet-Edgar integral representation theorem, and the Ryll-Nardzewski fixed point theorem. The amart theory led to the following result that can be stated without amarts:

Preface

xi

scalar convergence of stochastic processes implies weak a.s. convergence if and only if the dual Banach space has the Radon-Nikodym property. We

regret that this book does not include the important characterization of unconditionality of Lr bases and convergence of martingale transforms in terms of Burkholder's biconvexity-space limitations precluded it. Chapter 6 deals with martingales: maximal inequalities, LP laws of large numbers for martingale differences (extended to a class of mixing processes),

decompositions, convergence of transforms of martingales, Burkholder's square-function inequalities. The Maharam lifting theorem is derived from the martingale theorem. In Chapter 8 we derive Ll ergodic theorems (the general Chacon-Ornstein theorem, the Chacon theorem) from the superadditive ratio ergodic

theorem. The identification of the limit by this method is particularly convenient.

Chapter 9 provides a second instance of the unity of ergodic and martingale theory in our treatment. A general principle is proved that shows how multiparameter convergence theorems may follow from one-parameter maximal and convergence theorems, provided that the function space has

the property called "order continuous norm"-the Lp spaces have it, and Orlicz spaces have it under (02). Hearts of Orlicz spaces always have order continuous norm. We believe that nearly all the known multiparameter ergodic and martingale theorems with convergence to infinity over

"rectangles"-the indices converge to infinity independently-can be obtained by this method. Multiparameter maximal theorems are not needed anymore for convergence.

This is a research book in that most of the content has previously been published only in research articles, and there is some novelty in presentation. On the other hand, we believe that our approach is particularly simple and economical and therefore the book is of some pedagogical interest. In a graduate course taught to students with knowledge of measure theory only, we would cover Chapter 1, some basic results from Chapter 2, most of Chapters 3 and 6, essential convergence of martingales from the beginning of Chapter 4, derivation from the beginning of Chapter 7, and the conservative Chacon-Ornstein theorem from Chapter 8. The book could also be used for a more advanced topics course in probability, ergodic theory (Chapters 8 and 9), derivation (Chapters 4 and 7), or Banach spaces

(Chapter 5). Any such course would begin with Chapter 1 and the main results from Chapters 2 and 3. We are grateful to our many colleagues, collaborators, and students whose work appears here: especially Mustafa Akcoglu, Ken Astbury, Don Austin, S. N. Bagchi, Alexandra Bellow, Antoine Brunel, R. V. Chacon, Bong Dae Choi, Leo Egghe, Hans Follmer, Jean-Pierre Fouque, Nikos Frangos, Nassif Ghoussoub, Ulrich Krengel, Michael Lin, Annie Millet, Laszlo Szab6, Michel Talagrand, and Zhongde Yan. Thanks are due to the National Science Foundation for support of some of the research that now

xii

Preface

appears in the book. We wish to thank the Department of Mathematics of Ohio State University for bringing us together in the first place, and providing the atmosphere that made the book possible. Thanks are also due to Cambridge University Press, especially editors David Tranah and Lauren Cowles, for their patience and help. Finally, we wish to express our deep gratitude to Annie Millet, who has-as the reader for the Cambridge University Press-corrected many errors and made suggestions for improvements.

Columbus, Ohio January, 1992

G. A. Edgar Louis Sucheston

1

Stopping times

We will begin with the material that will be used throughout the book. The idea of the stopping time, especially the simple stopping time, is central. The setting in which this naturally occurs involves Moore-Smith convergence, or convergence of nets or generalized sequences. This will be useful even if we are interested only in sequences of real-valued random variables; but will be even more useful when we consider derivation (Chapter 7) and processes indexed by directed sets (Chapter 4). Given a stochastic process (Xn), a stopping time is a random variable

T taking values in IN U loo} such that, for each k, the event Jr = k} is determined by the first k random variables X1, X2, , Xk. A process (Xn) is an amart if for every increasing sequence Tn of bounded stopping times, E [XTn] converges. (For variants of this definition, see Section 1.2.) The main result of this chapter is the amart convergence theorem for the index set IN, proved in Section 1.2. The argument, using stopping times, is elementary, and may be followed by a reader with only a basic knowledge of the measure theory. To make the point, we will sketch the proof of almost

sure convergence of an amart (Xn) with integrable supremum. The basic observation is that there is an increasing sequence Tn of simple stopping times such that XTn converges in probability to X * = lim sup Xn. The reason for this is that lim sup-or any other accumulation point-manifests itself infinitely often on the way to infinity; it is like a light shining on the horizon. (This is not true of the supremum, which can all too easily be missed.) Thus, after we obtain X, close in probability to X*, we can find r2 > Tl such that XT2 is even closer to X*, etc. See (1.2.4). Similarly, we obtain an increasing sequence an of simple stopping times such that Xon converges in probability to X. = lim inf Xn. Integrating XTn and Xon

and using the amart property, we can conclude that X* = X. The a.s. convergence of L1-bounded amarts follows by truncation or stopping: The class of L1-bounded amarts is closed under both operations. First we prove a general principle (1.1.3), showing how in metric spaces

the behavior of sequences determines that of nets. From the amart convergence theorem, using (1.1.3), we easily obtain stochastic convergence of Ll-bounded ordered amarts indexed by directed sets, which in turn implies Krickeberg's theorem that Ll-bounded submartingales converge stochastically. Quasimartingales are defined on directed sets, and are also shown

to be ordered amarts, so that they also converge stochastically, andunder proper assumptions-essentially. To understand the difference between martingales, submartingales, quasimartingales, and amarts, assume

Stopping times

2

the index set in IN, and consider a deterministic process (12 has only one element). Then amarts are exactly convergent sequences of numbers; thus it is not surprising that also in general amarts will be sufficient for convergence,

and for the existence of limits in the continuous parameter case (Edgar & Sucheston [1976b]). A deterministic quasimartingale is a sequence (or function) of bounded variation; the analogy of the Lebesgue-Stieltjes integral rightly indicates that the full notion of quasimartingale will be required for stochastic integration. The sequential sufficiency theorem (1.1.3) is especially adapted to the needs of this book, first because it does not require convergence of all increasing sequences, and second because it does not assume that the metric space is complete, permitting in (5.3.37) a sequential proof of convergence in Pettis norm of amarts with values in a Banach space. Another important idea of the book is derivation. As an illustration of a derivation theorem, Section 1.3 contains a proof of the Radon-Nikodym theorem. Other derivation theorems will be seen later in the book. Section 1.4 discusses the conditional expectation. After it has been defined, the conventional definition for martingale can be seen to be equivalent to the definition in terms of stopping times.

I.I. Definitions Directed sets

We begin with some of the basic definitions of our subject. The first goal is the discussion of stopping times. As a preliminary to that we consider Moore-Smith convergence.

(1.1.1) Definition. A directed set is a nonempty set J, together with a binary relation < satisfying: (1) if tl < t2 and t2 < t3i then tl < t3;

(2) t 0, there exists to E J such that p(at, a) < e for all t > to. Notation: lim at = a; tEJ

the "direction" in J is understood. A few of the elementary properties of nets are stated in the Complements,

below. However, a few facts that will be used often are proved here.

A net (at)tEJ in a metric space (S, p) is a Cauchy net if, for every e > 0, there exists to E J such that for all t > to we have p(at, ato) < e. Every convergent net is a Cauchy net. In general, the converse is false: in the metric space of rational numbers, even some Cauchy sequences fail to converge. A metric space S is called complete if every Cauchy sequence converges in S. In fact, this implies also that every Cauchy net converges in S. This is part of the next result:

(1.1.3) Sequential sufficiency theorem. Let S be a metric space with metric p and let (at) be a net in S. Suppose either: (1) (at,,) converges in S for every increasing sequence (tn) in J; or (2) there exist indices sn. E J such that the sequence (at )nEJN converges

in S for every increasing sequence (tn) in J with to > sn for all n; or (3) (at) is a Cauchy net and S is complete. Then the net (at) converges.

Stopping times

4

Proof. Hypothesis (1) implies hypothesis (2), so case (1) of the theorem follows from case (2). It is therefore enough to prove cases (2) and (3). We show now that if (at) satisfies (2), then it is a Cauchy net. Indeed,

if (at) is not a Cauchy net, then there exists s > 0 so that for every to E J, there exists t > to with p(at(,, at) > 2e. We show there are indices to > so, tl > sl such that p(ata, at1) > e. Indeed, there is t' with t' > to and ti > si. If p(ato, ati) > e, set t1 = ti and we are done. If not, p(ato, at,,) t' (hence tl > Si) such that p(at1, at,) > 2e. Then p(ato, at1) >_ p(atl, ati) - p(ato, at1) > 2e - e = E. Since the index set J is directed, there exists an element tl E J such that t1 > to, tl > sl, and p(ato, at1) > E. This can be applied recursively to define a sequence

tl < t2
sn and p(at,,, ats}1) > e for all n, contradicting

Now suppose (at) is a net satisfying either (2) or (3). Since (at) is a Cauchy net, we may construct recursively a sequence tl < t2 < such that for all t > tn, we have p(atn, at) < 2-n. In case (2), again since J is directed, we may choose to so that in addition tn > sn. Now (at,)nEIN converges in S; in case (2) this is true by hypothesis, and in case (3) since (at,) is a Cauchy sequence and S is complete. Write a for the limit of the sequence (atn). We claim that, in fact, the entire net (at) converges to a. To see this, note that p(atm, atn) < 2-` for all n > m; so p(atm, a) < p(atm, atn) + p(atn, a) < 2-m + p(atn, a). The last term can be made as small as we like, so p(atm, a) < 2-'. Thus, for all t > t,n, we have p(a, at) p(a, at_) + p(atm, at) < 2-- + 2-- = 2-m+i. Stochastic basis

Let (SZ, F, P) be a probability space; that is: S2 is a nonempty set, F is a or-algebra of subsets of St, and P is a countably additive, non-negative measure on F with P(Q) = 1. A sequence (Fn)nEIN of sub-a-algebras of F is a stochastic basis if it satisfies the monotonicity condition: if m < n,

then Fm C F. A probabilist refers to f as the sample space, the elements of F as events, and the number P(A) as the probability of the event A. In this terminology,

the index set IN can be considered to be "time," and fn is the collection of events "prior to time n" (in the wide sense: < n). For an event A E Fn, if we are given everything that is known at time n, then we should be able to determine whether or not event A has occurred. The data (1, F, P) and (.Fn)nEIN will be fixed throughout most of the book. Suppose a stochastic basis (Fn) is given. A sequence (Xn)nEyq of random variables is said to be adapted to the stochastic basis if Xn is Fn-measurable

for all n E IN. Such an adapted sequence (Xn) of (real-valued, finite) random variables may be called a stochastic process indexed by IN.

If a sequence (Xn) of random variables is given, but not a stochastic basis, there is a canonical way to construct a stochastic basis: the a-algebra .Fn should be the least a-algebra of subsets of f so that the random variables , Xn) for this X1i X2, , Xn are measurable. We write Fn = a(Xl, X2, a-algebra.

1.1. Definitions

5

Stopping times A function a: S2 -> IN U {oo} is a stopping time [for the stochastic basis (.Fn)] if it satisfies

{a=n}E.Pn

forallnEN.

An equivalent definition is:

{a E [IXTI] > E [IXTI IAN]

AP(AN).

Now as N -> oo, the set AN increases to {supra All > A}, so the result follows by the monotone convergence theorem. Complements

(1.1.8) (Uniqueness of limits.) Let (at) be a net in a metric space S. Then (at) converges to at most one point of S. (1.1.9) (Subnets.) Let J and L be two directed sets. A function cp: L -> J is called cofinal if, for every t E J there exists so c L such that cp(s) > t for all s > so. If (at)tEJ is a net in S, then the net (b.)-EL is a subnet of (at) if there is a cofinal function cp: L -+ J such that b8 = a,(8) for all s E L. Prove: if (at) converges to a E S, then the subnet (be) also converges to a. (1.1.10) (Frechet property.) Let (at) be a net in a topological space S. Let a E S. Suppose that every subnet (b8) of (at) admits a further subnet (ca) that converges to a. Then (at) itself converges to a. (See, for example, Kelley [1955], Chapter 2.) (1.1.11) (Boundedness.) bounded.

A convergent real-valued net need not be

(1.1.12) (Measurable approximation lemma.) Let (Fn) be a stochastic basis. Write .F,,,) for the a-algebra generated by the union UnEIN .F,2. Let Y

be an F,-measurable random variable. Then for every e > 0, there exist an integer n E IN and a random variable X, measurable with respect to .fin, such that P j IX - Y J > -I < e. To prove this, observe successively that the set of random variables Y with this approximation property: (a) is a linear space; (b) is closed under pointwise a.e. convergence; (c) contains indicator functions 1B for B E Fn; (d) contains 1B for B E (e) contains all

random variables measurable with respect to F. (1.1.13) (Optional stopping of the stochastic basis.) If (Fn) is a stochastic basis, and T E E, then F, is a a-algebra.

Stopping times

8

(1.1.14) (Monotonicity of stopping.) If a < r, then F, C Fr. (1.1.15) (Generalized waiting lemma.) Let a E E be given, and for each

m E IN with {a = m} # 0, let ri'i E E be given with T(m) > in on {a = m}. Then T defined by r(w) = T(m) (w) on {a = m} belongs to E and

T> a. (1.1.16) (Weak L1.) A random variable Y is said to belong to "weak L1" if there is a constant C such that P{IYI > Al < C/A for all A > 0. Prove that all L1 functions belong to weak L1, but that the converse is false.

(1.1.17) (Maximal weak L1.) Suppose (Xn) is a stochastic process such that supE [IXQI] < oo.

oEE

Then (by the maximal inequality 1.1.7) the corresponding maximal function

sup, IX,,l belongs to weak L1. Show that the maximal function need not belong to L1.

(1.1.18) (Reversed stochastic basis.) A reversed stochastic basis on (Q, F, P) is a family (.Fn)nE-IN of a-algebras satisfying the monotonicity condition

F

-F- 1

2 F 3 J ...

Simple stopping times are defined as before; we will continue to write E for the set of all simple stopping times. (It is now a dual directed set.)

Prove the maximal inequality in this setting: Let (X,)nE_IN be a sequence of random variables adapted to the reversed stochastic basis Then for every A > 0,

P{ sup IXnI>A}< 1supE[IXQI] nE-IN

J

A 1EE

(Edgar & Sucheston [1976a]). Remarks

Moore-Smith convergence was proposed by Moore [1915] and Moore & Smith Its usefulness in general topology was displayed by Birkhoff [1937], and carried out in detail by Kelley [1950]. A particular case of the sequential sufficiency theorem 1.1.3 was proved by Neveu [1975]: A family indexed by a directed set converges in a complete metric space if there is convergence for all increasing [1922].

sequences.

The importance of stopping times was emphasized by Doob [1953]. The maximal inequality 1.1.7 is from Chacon & Sucheston [1975].

1.2. The amart convergence theorem

9

1.2. The amart convergence theorem Let a probability space P) and a stochastic basis (.F) be fixed. We will write, as before, E for the set of all simple stopping times for If (X,,,) is a stochastic process, and a E E, then the stopped random variable XQ makes sense. If X,, is integrable for all n, then of course Xo is integrable, since a has only finitely many values. An adapted sequence (Xn) of integrable random variables is called an amart if the net (E [Xe] )oEE of real numbers converges. That is, there is

a real number a with the property: for every e > 0, there exists vo E E such that, for all or E E with a > ao, we have

IE[Xc] - aI < e. Since the metric for the real line is complete, this is equivalent to a Cauchy condition: for every e > 0, there exists ao E E such that, for all v, T > ao, we have IE[XX] - E[XT] I <E. The definition may, in turn, be phrased in terms of sequences only: The

integrable stochastic process (Xn) is an amart if and only if, for every sequence vl < a2 < in E with vn -* oe, the sequence (E [Xon] )71EIIv converges.

If the net E [X.,] is constant, then of course it converges. An adapted sequence X,, with this property is called a martingale. Some other interesting examples of amarts are given below; see, for example, (1.4.4). Observe that in the definition of amarts it is essential to use simple stopping times: otherwise the famous original gambling martingale, in which

the player doubles his stakes each time he loses, would not be an amart (see Section 3.2).

The lattice property

The collection of all Li-bounded amarts satisfies a very useful lattice property: if (X,,) and (Y,,) are both Ll-bounded amarts, then so are the pointwise maximum (X,a V Y,,) and the pointwise minimum (X,,, A YY,). This will be proved below (1.2.2). A similar result is true for semiamarts. A stochastic process (Xn) is a semiamart if supQEE IE [Xe] I < oo.

(1.2.1) Lemma. Let (X,,) be an amart. Then (X,,,) is a semiamart. Proof. The net E [X , j converges, so it is a Cauchy net. There is N E IN such that IE [XN] - E [XQ] I < 1 for all a E E with a > N. If a is any simple stopping time, then or V N is a simple stopping time > N. But IE [XonN] I < E Amax jXnj]

and IE [XovN - E [XN] ] I < 1, so IE [X,,] I = IEr[XaAN] + E1[XXVN] - E [XN]

<E r 1max jXnj +1 X v - X,f . Taking expectation, we have (1.2.2a)

E [X,,] -E [XT] <E[Xp] -E[X.,].

To prove (1), let a E E be arbitrary, and choose r > a constant, r = m. Then E [XQ ] < E [Xp] + E [X,-] < sup,,,EE E [X.] + SupnEIN E [IXnI]

To prove (2), choose Qn E E, an -> oo, such that

E [X n ] - lim sup E [X, ] a

and then choose

Qn such that E [X n ] -> lim inf E [X,, ] a

.

Since (Xn) is an amart, we have E [X4] - E [X n ] - 0 by inequality (1.2.2a). Therefore lim sup E [X,+] = lim inf E [X,+], possibly both infinite. For (3), combine (1) and (2). For (4), observe that x V y = (x - y) + + y and x A y = x - (x - y)+; then apply (1). For (5), apply (3), using the same identities.

(1.2.3) Corollary. Let (Xf)fEnv be an Ll-bounded amart. (a) If A is a positive constant, then the truncation ((-A) V Xn A A)nEIN is also an amart. (b) SupaEE E [IX.,, 1] < oo. (C) SupnEIN IXnI < 00 a.s.

Proof. For (a), apply "the lattice property," that is, part (5) of the theorem. For (b), observe that I xI = 2x+ -. x. For (c), apply the maximal inequality (1.1.7) using (b).

1.2. The amart convergence theorem

11

Convergence

We are ready now for the first convergence theorem. We begin with a useful observation on approximation of cluster points by stopping times.

Cluster point approximation theorem. Let (Xn) be an

(1.2.4)

adapted stochastic process, let Y, = a(UnE1N fn), and let Y be an F,,,,-measurable random variable. Suppose that Y(w) is a cluster point of the sequence

for every w E Q. Then there exists a sequence

al < a2 < in E such that an --> oo and limn-w Xo = Y a.s. Proof. Given any N E IN and e > 0, we will construct a stopping time or E E with a > N and P{JXa -YI < e} > 1-e. This may then be applied recursively to produce an increasing sequence an E IN with Qn -+ oo such that XQ converges to Y stochastically (that is, "in probability"). Then there is a subsequence that converges a.s.

First, since Y is .F -measurable, by (1.1.12) there is N' > N and an .2N'-measurable random variable Y' such that

P{lY - Y'l < 2} > i

(1.2.4a)

2

But { rY

- Y'l < } C {there exists n > N' such that lXn - Y'l < 21' 2

Therefore, there is an integer N" > N' such that P(B) > 1 - e/2, where

B = {there exists n with N' < n < N" such that IXn - Y'I < 21' Define the simple stopping time a as follows:

a(w) =

J inf{n:N'm is an amart. (1.2.13) (Associated charge.) Let (X,,) be an amart. Then for each A E U,,F,,, the limit p(A) = lim E [Xn 1A] n-oo exists and p is a finitely additive set function. Remarks

The term "amart" comes from "asymptotic martingale." Theorem 1.2.4 on approximation of limit points is from Austin, Edgar, Ionescu Tulcea [1974]. The lattice property was given explicitly in Edgar & Sucheston [1976a], but implicitly already in Austin, Edgar, Ionescu Tulcea [1974].

The amart convergence theorem was stated explicitly in Austin, Edgar, Ionescu-Tulcea [1974]. The key element of the theorem is the use of simple stopping times. Their proof used the method of "up-crossings." The proof given here, based on truncation, is from Edgar & Sucheston [1976a]. Earlier versions of this theorem and related theorems are found in Baxter [1974], Lamb [1973], Mertens [1972], Meyer [1966].

1.3. Directed processes and the Radon-Nikodym theorem We will see many "derivation theorems" in this book. In this section we will prove the Radon-Nikodym theorem as an elementary example. For this purpose (and for its usefulness in the future), we discuss processes with

index set more general than V. In Chapter 4 we will discuss more thoroughly the theory of amarts, ordered amarts, and other processes, indexed by a directed set. But some basic results will be proved here. Processes indexed by directed sets

Let (St, F, P) be a probability space, and let J be a directed set. The family (.Ft)tEJ of a-algebras contained in .F is a stochastic basis if it satisfies

the monotonicity condition: F. C_ Ft whenever s < t in J. The family (Xt)tEJ of random variables is adapted to (.Ft) if, for all t E J, the random variable Xt is Ft-measurable. A function a : S2 -+ J (a J-valued random variable) is a simple stopping time for (.Ft) if o, has finitely many values and {a = t} E Ft for all t E J. (Because a has finitely many values, this measurability condition is equivalent to: for < t} E Ft for all t.) We will write E((Ft)tEJ) or E for the set of all simple stopping times for (.Ft). It is a directed set itself as before. An smart for (.Ft) is an adapted family (Xt) of integrable random variables such that the net (E [Xa] )aEE converges. An ordered stopping time is a simple stopping time r such that the elements t1, t2, , in the range of rr are linearly ordered, say t1 < t2 < < tm. We denote by E° the set of ordered stopping times. Then E°

Stopping times

14

is a directed set under e}=0

tEJ

for every e > 0; and converges in mean if limt E [IXt - X" I] = 0. Stochastic convergence is determined by a metric, for example

p(X, Y) = E [IX - YI A 1]

,

or p(X, Y) = E [1

Y

I+

X

]

(These equivalent metrics are complete, but we do not need that fact here.) Recall that a family (Xt)tEJ of random variables is uniformly integrable if

lim supE [IXtI 1{1xtI>a}] = 0.

A-'OO tEJ

(1.3.1) Theorem. Let J be a directed set, and let (.F't)tEJ be a stochastic basis. Let (Xt)tEJ be an ordered amart (or amart). If (Xt) is L1-bounded, then it converges stochastically; if (Xt) is uniformly integrable, then it converges in mean.

Proof. By the sequential sufficiency theorem (1.1.3), it is enough to show that there exist sn E J such that converges stochastically for

each increasing sequence t1 < t2 < ... in J with tn, > sn. Choose sn, increasing so that IE [XT, - XT2] I < 2-' for r1, r2 E Eo, T1,72 > S.Suppose t,, increases and to > sn. We claim that (Xt.) is an amart for the stochastic basis (Gn) defined by 9,, = .Ft.. Given e > 0, choose N with 2-N < e; for a E E((cn)) with a > N, we have tQ > tN > SN. Also tQ E E((.Ft)), since {v = s} E G. = .Ft. and or = s implies to = t87 so the sets It, = t8} are countable unions of sets in Ft, = G8, and hence is are in G8. Then IE [Xto - XtN] I < 2-N < E. Thus (E [Xto]

1.3. Directed processes

15

Cauchy. Thus (Xt.) is an amart. It therefore converges a.s. by the amart convergence theorem (1.2.5), and thus it converges stochastically. On a uniformly integrable set, stochastic convergence coincides with convergence in mean. (See Theorem (2.3.4).) The question of pointwise convergence of (Xt) will be considered below in Chapter 4. This amart convergence theorem will be used to prove the Radon-Nikodym theorem. Notice that the Radon-Nikodym theorem has not been used up to this point-we have not even mentioned conditional expectation. Let (SZ,.F, P) be a probability space. A set function p: F -+ 1R is countably additive if, for any sequence (An) C F of pairwise disjoint sets,

it U An = > p(An)n=1

n=1

We say that p is absolutely continuous (with respect to P) if p(A) = 0 whenever P(A) = 0. The variation of p on A E F is n

Ip1(A) = sup

lµ(C2)I, i=1

where the supremum is taken over all finite pairwise disjoint sequences

(C1) 1cFwith Czc_A.

Our proof uses some facts about signed measures. They are proved in the Complements (below) for completeness, and so that it can be verified that they do not use the Radon-Nikodym theorem. The facts are: (1) Let p: F -> IR be a countably additive set function. If An E F and Al D A2 D , then , then µ((l An) = lim p(An). If Al C A2 C_ µ(U An) = lim p(An) (1.3.4). (2) A real-valued countably additive set function p has finite variation (1.3.5).

(3) Let p be a countably additive set function that is absolutely continuous with respect to P. Then for every e > 0, there exists 6 > 0

such that ifAEFandP(A) 0, pairwise disjoint [if A1, A2 E t, and Al 54 A2, then

Stopping times

16

Al n A2 = 0], and having union Q. Write J for the set of all measurable partitions. If s, t E J, then we say that t essentially refines s, and write t > s, if for each B E t there exists some A E s with B C_ A a.s. With this ordering, J is a directed set. If t E J, let Ft be the o-algebra generated by t; thus Ft is a finite algebra and t is the set of atoms of F. Clearly, if s < tin J, then F8 C _Ft. Next, for t E J, define a random variable

Xt =

P(A)1A. AEt

First we claim that (Xt) is an amart. [In fact it is a martingale: that is, E [X°] is constant.] Suppose o: S2 -+ J is a simple stopping time. Then for = t} E .fit for each t, so for = t} is a union of sets A E t. So

X° = tEJ

Xt 1 {°=t} = E Y, µ(A) 1A 1{°=t} = E E µ(A) P(A)1A' t

AEt

P(A)

t

AEt

AC{o=t}

Thus

E [X] _t E AEt E

u(A)

P(A)

AC {°=t}

=It

UA AEt

AC{°=t}

_ E µ{o = t} = 04 t

This shows that the net (E [X°]) is constant, and therefore convergent. So (Xt) is an amart. Next we must show that (Xt) is L1-bounded. Let t E J. Then

E [IXtI] = E

Ip(A>I

- P(A) < Iµi(S2)

AEt

But µ has finite variation, by (1.3.5), so (Xt) is Ll-bounded. We claim next that (Xt)tEJ is uniformly integrable. Let e > 0. Then, since µ is absolutely continuous, by (1.3.6) there exists 5 > 0 such that It(A)I < e whenever P(A) < 5. Let A = IpI(S2)/8. For all t E J,

P(A) P(A) = p

E [Xt 1{xt>a}] = AEt

µ(A)>AP(A)

U AEt µ(A)>AP(A)

A

AP(A)

Similarly

E [IXtI 1{x, to, then

E [Xt 1Ao] _ E IL(A) = p(Ao) AEt

ACAo

Thus IE [Y lAo] - IL(Ao) I < lim IE [Y 1Ao] - E [Xt 1Ao] I + lim IE [Xt 1Ao] - p(Ao)I

X. on the set {v = s}. Suppose that T takes values t1 < t2 < < t, . For a fixed value s of v, we use induction on the index n between 1 and m defined by to = max { T(w) : Q(w) = s }. Since or < T,

certainly to > s. For tn = s, we have r = s on for = s}, and therefore Efe [XT] = X8 on {v = s}. For the inductive step, suppose t,ti > s, and define T' E EO by

T'(w) =

T(w)

if o(w) = s and r(w) < t,,

t,i_1

if a(w) = s and r(w) = t,,,

T(w)

if or(w)

s.

Stopping times

22

Then r > T' > Q. By the induction hypothesis, E''- [XT-] > X. on to, = s}. Also, {T' s, on to, = s}, we have E-r' [XT] = E-F° [E'"Vtn-1 [XT]] > ET- [XT,] > X8.

This shows ET° [XT] > Xo and completes the proof of [XT] > X, for a, T E E0, a < T. On integrating, we obtain E [XT] = E [E- [X] ] > E [Xs] for or < T. Next suppose that the net (E [XT] )TEEo is increasing. We must show

-

-

that Ewe [Xt] > Xe if s < t. Let A E F. Then if wVA if wEA

s

{t

is an ordered simple stopping time, and or > s. Therefore E [Xo] > E [X8]. That is: E [Xt 1A] + E [X810\A] > E [X81A] + E [X810\A]

.

Therefore E [Xt 1A] > E [X81A]. This is true for all A CF8, so EF° [Xt] >_ X8,

as required. (ii) Since (-Xt) is a submartingale if and only if (Xt) is a supermartingale, (ii) follows from (i).

(iii) Suppose (Xt) is a martingale. Let a E E be given. Choose t > a. Then

E [X"] = E E [Xs 1{0=8}] 8 M - e.

i=1

Now let T E E° with r > sm,. Write tl < t2 < ... < to for the set of values of T. Now if we apply the inequality (1.4.4a) with the finite sequence subtract (1.4.4b), we obtain s,n, we have IE [Xrl] - E [Xr2] < 2e. Therefore (Xt) is an ordered amart. Riesz decomposition The Riesz decomposition (1.4.6) for an amart or an ordered amart shows that the process is "close" to a martingale. First consider an alternative ordering for the set EO of ordered stopping times. If a, ,r E E0, we write a < T iff there is t E J with v < t and t _< T. Note that EO is directed under the order as well as under the usual

order 0 there is v E E° such that l ar - al < e for all T E E° with a T. But there is t E J with t > a, and therefore l aT - aI < e for all r > t. Thus ar converges to a according to 0; choose s E J such that a > s, r > s and a, -r E E° implies E [XQ] - E [XT] l < E.

Lets < or 4 E. [Xn] for m < n, and a martingale by X. = Em [Xn] for m < n. It is easy to see by induction that these properties need to be checked only for n = m + 1. A quasimartingale is an integrable process (Xn) for which there is a constant C such that n

IIEI [Xi+i] - XiIIi < C for all n. Equivalently (1.4.30) there is a constant C such that

r,-i IE[XTt}1-XT,]I ,, fi2 and fii >-o fi2, then Le1(52, F, p) C L42 (52,F,µ). If Le1([0, oo)) C Lt2 ([0, oo)), then fii >-oo fi2 and fil >o fi2 (2) Suppose p is finite. If fii >- m fi2, then

L..1p) c Lee(I,F,li) If L41([0,1]) C L4.2 ([0,1]), then fii r,, fi2. (3) Suppose p has no arbitrarily small sets and fi2 is finite. If fit -o fi2, then L4,1(52, F, µ) C L.,2 (S2, If l4, C 14,2, then fi1 }-o 'b2Proof. (1) Suppose fil >-o fi2 and fii >-

b'fii(a'u) > fi2(u) b"fii(a"u) > fi2(u)

fi2. Thus

for u > u' for u < u".

We may assume fi2(u') < 00. If not, choose u < u' with 4D(u) < oo, and let a = a'u'/u, so that for u > u,

-b2(u) < fi2(u'u/u) < b4i(a u u/u) = b'fil(au). Similarly, we may assume fii(a"u") > 0. If we take a = max{a', a"},

b =max

fi2 (u)

fii (au)

u" < u < u'

,

2.2. More on Orlicz spaces

53

and b = max{b', b", b"',1}, then we have

b'11(au) > 't2(u)

for all u.

Let f E L4,1. Then M4,1(f /k) < 1 for some k. Now

M4,2(

k)=f'P2(Ia l) dµ

f bbl (ICI) dp 0, then 2nun/a > un for n larger than some no, so (2n un

f 2

a

2-n

4)1(2nun)

a

-n

00

E 'D2(un) It1(2nun) n=no 00

> E 1=00. n=no

Thus f ¢ Lt.2. (2) Now 4>1 >,, '102i say b-Pl(au) > uo. As before, we may assume 4D2 (uo) < 00. If f E L4,1, then M4>1 (f /k) < 00 for some k. Then

M4,2(j) = J P2(ka) da -w 42 fails, then the points u,, as in case (1) may be chosen with u" --> oo, so also 4;1(2"u") -> oo, and we may assume 411(2"u") > 1. Then the intervals A", chosen as before, have total length at most 1, so they may be chosen in [0, 1]. (3) Now 4i1 }o'D2i say b4i1(au) > 4i2(u) for u _< uo. As in case (1), we may assume 4i1(auo) > 0. If f E L4,1, then M,1(f /k) < oo for some k. Now

f

-i1

(iii)

dp > 4i1(auo)p{IfI >_ akuo},

so µ{I If I > akuo} < oo. Since µ has no arbitrarily small sets, we see that the set { I f I > akuo } consists of a finite number of atoms, so 4?2 (I f I /ka) is

bounded on the set. Now

MD, ( 'fa) = k

f

t2 1

I aI

I dµ

< f1JfJ- 4i1 i then we will say that 4i1 and 4D2 are equivalent, and write

(D1 ^ qt2

Largest and smallest Orlicz functions Consider the following conjugate pair of Orlicz functions (see Figures (2.2.4a) and (2.2.4b)). 0

OFmin(u)

{u u 00

1

0 4D(u), and if 0 < u < ul, then by conu > ul, we have vexity of 4D, we have iD(u)
a} > 0, then we would have 1 > f Pmax(If I /a) dµ > f{I fl>a} oo dµ = oo, which is false. So f E L,, and IlfIIoo < Now suppose f E Write a =IIf 114,m,.. Define

f=

f

if IfI a 0

fi=f - fog= (IfI -a) sgn(f)

if IfI a.

Then IIffII. < a, and IIfilli=

f Iflldi {IfI>a}

(IfI - a)dµ IfI - 1

=a

-IfI/a>1}

=a

f

dµ

a

{f

min (I fa I)

dµ < a.

Thus IIfiIii V IIf.IIpp -(, (D2 is: There exist a, b, no > 0 such that (D2(u)

for all u > no.

The following are equivalent: (a) (b1 >(b) there exist a,uo > 0 such that 4P1(au) > (u) for all u > uo. A similar equivalence holds at 0. (2.2.20) (Definition of Find Orlicz functions showing that the fol-

lowing are not equivalent: (a) 4P1 >-,, (D2i (b) there exist b, uo > 0 such that b41(u) > 4 2(U) for all u > uo. (2.2.21) (Convolution on 1Rn.) Let SZ be n-dimensional Euclidean space IRn and let p be n-dimensional Lebesgue measure. Let (D and T be conjugate Orlicz functions. Suppose (D satisfies the (A2) condition. If f E L,p(1R') and g E Lq,(IRn), then the convolution

k(t) = J n f (t - x)g(x) dx = fiRn f (x)g(t - x) dx exists and is continuous on IRn. If T also satisfies the (02) condition, then limlltll-,,. k(t) = 0 (Zaanen [1983], p. 600). (2.2.22) Let 11 be n-dimensional Euclidean space IRn and let p be ndimensional Lebesgue measure. Suppose 4 satisfies the (O2) condition.

If f E L. (1Rn), then, for any e > 0, there exists a continuous function g : IRn - > IR with compact support such that 11f - g

< e. For h E IRn, let the translate fh be defined by fh(x) = f (x - h). Then

Ilfh - f II4, - 0 as IIhII -' 0 (Zaanen [1983], p. 599). (2.2.23) Let A be a subset of H4,(IRn). Then A is relatively compact if and only if (1) SUPfEA IIhII(b < oo; and

(2) for every e > 0 there is 6 > 0 such that for all h E IRn with IIhII < 6 and all f E A, we have Il fh - f II 4, < e (Krasnosel'skii & Rutickii [1961], p. 100).

68

Infinite measure and Orlicz spaces

(2.2.24) (Linear functionals.) Let 4) be a finite Orlicz function with conjugate T. Let a be a (possibly discontinuous) linear functional on L,. Suppose

(a) a(f) < 1 for all f < 0; (b) there is e > 0 such that a(f) < 1 for all f with 1If III :5e. Then there is g E Lq,, g > 0 such that

a(f) = Jfgdµ for all f E

To see this, observe that by (a) a is a positive linear functional, and by (b) a is continuous (with norm < 1/e). Define a setfunction v on the sets A E F with u(A) < oo by v(A) = a(1A).

By (2.1.20) it follows that v is countably additive (on the ring of sets of finite p-measure), so by the Radon-Nikodym theorem (and the v-finiteness of p) there is g with a(A) = fAgdµ. The rest follows as in (2.2.11), using monotone convergence.

(2.2.25) (Finiteness.) It is possible for an Orlicz function ' to be infinite even if its conjugate 4 satisfies (02). For example ir X)

x

if0<x1. 1 arctan x (Thanks to A. Millet for providing this example.) Remarks

This section comes from the same sources as the last one: Krasnosel'skii & Rutickii [1961] discuss the comparison of Orlicz spaces in Chapter II, Section 13; and duality of Orlicz spaces in Section 14. Zaanen [1983], Chapter 19, also has a lot on duality. Fava [1972] introduced the spaces Rk, k = 0,1, in the context of ergodic theory. Frangos & Sucheston [1986] identified the spaces Rk, k > 1, as hearts of Orlicz spaces L logk L, and showed that Ra is a Banach lattice with order continuous norm. The identification of Ra as the heart of Ll + Lam, and the other material on the heart of the sum of Orlicz spaces, is from Edgar & Sucheston [1989].

2.3. Uniform integrability and conditional expectation Uniform integrability is a useful criterion in probability theory. We define

it here even for infinite measure spaces, so that the usual role played by Ll is taken by the largest Orlicz space Ll + L,,. Only the basic properties of uniform integrability are proved here, but it will reappear frequently throughout the book. An interesting characterization (2.3.5) of uniform integrability (which goes back to de La Vallee Poussin) can be formulated in terms of Orlicz functions. Then we will discuss conditional expectation on infinite measure spaces.

2.3. Uniform integrability

69

(2.3.1) Definition. Let IC be a family of real-valued measurable functions defined on (S2, , µ). We say that IC is uniformly integrable if, for every e > 0, there is A > 0 such that for all f E IC, we have f{IfI>a} If I dp < e. It would seem logical that a "uniformly integrable" set should consist of integrable functions. This is true if it is finite, but not in general. A singleton {g} satisfies this definition if and only if g E Ll + Lam, that is, g is in some Orlicz space Lp. Uniform integrability of a set JC might be considered as the property that the elements of IC belong to Ll + L,,,, in a uniform way. The main uses of the term "uniformly integrable" will occur when it is finite. Some books use a slightly different definition. We will say that IC is uniformly absolutely continuous if, for every E > 0, there is 6 > 0 such that if f E IC and A E F with µ(A) < 6, then fA IfI dp < e. The next result shows that this concept is (for most purposes) equivalent to uniform integrability. [The vector-valued versions are different, however; see (5.2.15).]

(2.3.2) Proposition. Let IC be a family of real-valued measurable functions defined on (Q, .F, µ). (1) If 1C is uniformly integrable, then IC is uniformly absolutely continuous. (2) Suppose IC is Ll + Lam-bounded. Then IC is uniformly integrable if and only if IC is uniformly absolutely continuous. (3) Suppose p is atomless. Then IC is uniformly integrable if and only if IC is uniformly absolutely continuous.

Proof. (1) Suppose IC is uniformly integrable. Let e > 0 be given. Then there is A so that for all f E IC, we have f{Ifl>a} I f I dp < e/2. Let 6 < E/2A. Now if f E IC and p(A) < 6, we have

f If I dp

fAn{IfI_a} I f I dµ + 2

fan{IfI_-}

If Idµ<M

for all f c K. Let e > 0 be given. Now there is 6 > 0 so that for µ(A) < 6 and f E IC we have fA If I dp < e. Let A > max{a, M/6}. Then µ{IfI

?A}a}

0 so that fA IfI dµ < e whenever f E K and p(A) < b. Let A = 2e/b. We claim that µ{IfI > A} < b for all f E K, which will prove that K is uniformly integrable. Suppose not: suppose that, for some f E K, we have µ{IfI > A} > S. Since p is atomless, there is a set

A C {IfI > Al with u(A) = 6/2. Then

>f IfIdµ>

2b= e,

A

a contradiction. There are some obvious sufficient conditions for uniform integrability.

(2.3.3) Proposition. Let K be a family of measurable functions. (a) Suppose there is f E Ll + L,,,, with IgI < If I a.e. for all g E K (we say K is dominated by f). Then K is uniformly integrable. (b) Suppose there is f E Ll + L,, with p{II > A} 5 tL{IfI > A} for all A > 0, g E K (we say K is dominated in distribution by f). Then K is uniformly integrable.

Proof. Part (a) is an easy consequence of (b); so we prove (b). Suppose

f = fi + foo, where f, c: Ll and f,, E L,,. Write M =

For

A> M, we have for aligE K, IgI dµ = Ap{IgI > Al + {IsI>_A}

f

µ{IgI >_ t} dt

_A}+ TO plIf I >t}dt

At{If1I+M>A}+f"o µ{If1I+M>t}dt

f

a Ifi I+M>a}

(If1I+M)dµ.

This is independent of g and tends to 0 as A -> oo.

One common use of uniform integrability is to connect convergence in mean with other modes of convergence. Recall that a sequence (fn) converges to f in measure if, for every e > 0, we have lim

n-,oo

µ{Ifn- fI >e}=0.

2.3. Uniform integrability

71

(2.3.4) Proposition. Let (Q, -'F, p) be a finite measure space. Suppose fn E Li for n E IN and f E Li. Then II fn - f IIi -> 0 if and only if { fn : n E IN } is uniformly integrable and fn -> f in measure. Proof. Suppose Ilfn

- f Iii - 0. Now first, if E > 0, then

i{Ifn - fl > E}
oo, so fn -+ f in measure. Second, we must prove uniform integrability. But { fn : n E IN} is Libounded, so by (2.3.2(2)) we may prove uniform absolute continuity. For

e > 0, there is N with Ilfn - flli < eforn>N. Now

9=imax IfnlVIfI is integrable, so there is 6 > 0 with

fl9ld/-L<e

whenever p(A) < 6.

Now for n < N, if p(A) < 6 we have

flfnldPflYldILN,if u(A) f in measure. Of course, K = If } U { f,, : n E IN } is also uniformly integrable. Fix E > 0.

There is 6 > 0 so that fA I9I dµ < e whenever g E K and p(A) < 6. Also,

there is Nso that if n>Nwe have µ{Ifn -fI>e}N,

f

Ifn - fldµE} = 2E + ep(SZ).

Thus IIfn - f Ill - 0.

(IfnI+IfI)du+

f

{I!n-J I 0, let A be such that 4D(u)/u > M/E for u > A/M. Then IfI d M {IfI_a}

IfIdµ =

µ

{IfI>_a} M

<MMJ

'1

((M)

µ<E.

{ISl?a}

So 1C is uniformly integrable. (1) (3). Suppose 1C is uniformly integrable. Write

C(A)=sup

. IfIdµ: f E J({ If 12 a) Then C is a decreasing function; lima.. C(A) = 0 since 1C is uniformly integrable. Choose A0 with C(Ao) < 1. Then - log C(A) is a positive

increasing function of A for A _> A0, and - log C(A) -> oo as A -+ oo. There

is a continuous function cp: (0, oo) -4 (0, oo) with cp(u) = 0 for u < A0, cp(u) < - log C(u) for u > A0, and limo-(,. cp(u) = oo. We may assume cp is strictly increasing on (A0, oo). Thus cp is an Orlicz derivative. Let 4 be the corresponding Orlicz function and let 0 be the inverse of V. Now -t(u)/u -> oo since W(u) -4 oo. Since cp is strictly increasing on (A0, oo), we

have cp(l,(y)) = y for ally > 0. Thus y < Now for f E K, we have

Me(f)

log C( (y)), or C(r(y)) < e-v.

<J(ifi) Ifidi V(If(wI)

in

oo

- Jo

o

dy If(w)I dµ(w)

{If(')I? '(v)}

If (w) I dµ(w) dy

oo

ff e_v

C(b(y)) dy

0


0, provided we allow EN, [f G] to take the value oo. (Compare (1.4.8).) Several of the usual properties may be checked easily: Eµ [h 1 + f2 191 _ EN. [ fl 191 + Eµ [ f2 19] (conditional expectation is a linear operator). If f > 0, then Eµ [f I G] > 0 (conditional expectation is a positive operator). If g is a-measurable, then E. [ g f 19] = gEµ [ f 19], provided both conditional expectations exist. Jensen's inequality is an important and useful property, which will be

used to prove that conditional expectation is a contraction on all Orlicz spaces.

(2.3.10) Jensen's inequality. Let f E (L1 + Lam) (1l, .F, µ), let 9 C .F be a v-algebra, and let -D: lR -> 1R be a convex function. Then

,D(El[.fI9])

<EN,[-D(f)I9]

a.e. on the set of v-finiteness of g.

Proof. First recall that a convex function on IR is necessarily continuous, so that 4(f) is a measurable function. Consider rational numbers m and b such that the graph of the line y = mx + b is below the graph y = (D(x); that is, 4D(x) > mx + b for all x E 1R. Then for any A E G with u(A) < oo, we have JA 4D(f) dp > m fA f dp + bµ (A).

Thus

L1ImJA11d1dp+bµ(A) or

j

(Eµ

[

(.f) I G] - mEµ [f I G] - b) dp > 0.

The integrand is a-measurable, and the inequality holds for all A E G with p(A) < oo, so Em

[4(.f)I9]-mEµ[.fI9]-b>0

Infinite measure and Orlicz spaces

76

y=mx+b Figure (2.3.10). Diagram for Jensen's inequality.

a.e. on the set B of o-finiteness of µ with respect to G. There are countably

many pairs (m, b) of rationals such that y = mx + b is below y = P(x). Thus there is a single set N E G with µ(N) = 0 so that, for all w E B \ N, Eµ [,P(f) I G] (w) > mEµ [f 19] (w) + b for all such pairs (m, b).

Now let WEB\N,ande>0. Write For each E > 0, there is a pair (m, b) of rationals with y = mx + b below y = -P(x), but yo < mxo + b + E. See Figure (2.3.10). (This is true since the convex function fi(x) is left-differentiable at xo.) Thus we have Eµ [-t(f) 19] (w) ? 4,(Eµ [f 191 (w)) - E. This is true for all e > 0, so we have Eµ

19] >,P(Eµ [f 19])

Note that if (P (0) = 0, then we even have EN a.e. on Q.

a.e. on B. I G] >

(Eµ [ f I G] )

(2.3.11) Corollary. Let 4D be an Orlicz function. If f E Li., then Em [f 9]EL4, and IIEµ [f 1G] II4, < If 11.t.

If f c Hp, then Eµ [ f 19] E H.. Proof. Both parts follow from: Mt (Eµ [ f I G]) Mt (f ). This is a consequence of Theorem (2.3.10), using the convex function 4(JxI).

2.3. Uniform integrability

77

(2.3.12) Corollary. Let K be a uniformly integrable set of F-measurable functions, and let { Gi : i E I} be a family of sub-c-algebras of F. Then

K={EM[fIGt] : f EIC,iEI} is also uniformly integrable.

Proof. By Theorem (2.3.5), there is an Orlicz function with 4?(u)/u -+ o0 such that K is L4,-bounded. By Corollary (2.3.11), the set k has the same

bound in L,. Complements

(2.3.13) (Uniform integrability and uniform absolute continuity.) Suppose

(1,F,µ) has an atom: A E F, 0 < µ(A) < oo, and if B C A, then either p(B) = µ(A) or p(B) = 0. Let fn = n IA. The set { fn : n E IN} is uniformly absolutely continuous, but not uniformly integrable.

(2.3.14) (The converse is false in (2.3.3(b)).) Let (52,.x', µ) be [0,1] with Lebesgue measure. For k > 3, let Jk = kl(o,1/(klogk))

Then the set K = { fk : k = 3,4,- } is uniformly integrable. But if K is dominated in distribution by f, that is µ{Ifkl _ A}-A} for all k, then we have f If I dt = oo, so f ¢ LI + L,,. = Ll (Clarke [1979]). (2.3.15) (Mean convergence of nets.) Let J be a directed set, and suppose

ft E Ll for each t E J. The net (ft) is said to be uniformly integrable at infinity if, for every e > 0, there exists s E J and A > 0 such that for all

t>s

LIft1>X}

IftIdµ<e.

Suppose µ(S2) < oo. Then: 11ft - f III -> 0 if and only if ft -* f in measure and (ft) is uniformly integrable at infinity. (Neveu [1965a], p. 54.) Uniform integrability is connected to weak compactness. The classical theorem along these lines deals with finite measure spaces: Let µ be a finite measure, and let K C L1(µ). Then K is relatively sequentially compact in

the weak topology of Ll if and only if K is uniformly integrable. (For example, Dunford & Schwartz [1958], (IV.8.11).) Here is the more general version.

Infinite measure and Orlicz spaces

78

(2.3.16) Proposition. Let 1C C L1 + L,,.. Then IC is relatively sequentially compact in the weak topology a(Li + Lam, L1 n Lam) if and only if 1C is uniformly integrable and L1 + Lam-bounded.

Proof. Suppose 1C is relatively sequentially compact. Then, for every measurable function h E L1 nL,,,, the set { f f h dp : f E 1C } is a bounded set of scalars. Thus, by the uniform boundedness principle, { If IIL1+Lo, : f E IC} is bounded. We claim that IC is uniformly integrable. Suppose not. Then there exist a>0,A, E.F,fnEIC with µ(An)-.0but Taking a 2-1 and subsequence, we may assume that µ(An) fn converges for the Now A = U' 1 An has weak topology v(Ll + Lam, L1 n L..), say fAnIfnIdp>a.

finite measure, and fn -> f weakly in L1(A). By the classical theorem, fAn I fnI dµ -+ 0 since µ(An) -r 0, a contradiction. Conversely, suppose IC is uniformly integrable and L1 + Lam-bounded. Now S2 is or-finite, say Ak I St, p(Ak) < oo. Let fn E 1C. For each k, the set { fn 1Ak : n E IN } is uniformly integrable and bounded in L1, so by the classical theorem, it is relatively weakly sequentially compact. We may piece together the limits, so there exists a measurable function g with

fn IA,, ` g 1& weakly as n -> oo, for all k. Now we claim that fn -> gin a(Li + Lam, L1 n Lam). Let e > 0. Let h E L1 n L. Choose A so that

{Ifn-91>a}

Ifn - gI dp
N, we have

JAk

- g)h dµ

< 3.

Then we have, for any n > N,

f

(fn - g)h dy

J (fn - g)hdp + k

+

I

Ifn-9I?a}\Ak

f

{Ifn-9I gin a(L1 +

L1 n L,,).

Ifn - gI I hl du

2.3. Uniform integrability

79

Another variant uses this definition: IC is uniformly Ro if for every 'y > 0

there exists M > 0 such that if A E .P and p(A) < y, then fA If dp < M for all f E K. Obviously a singleton If } is uniformly Ro if and only if f ERo. (2.3.17) Let K C Ro. Then K is weakly relatively sequentially compact in the weak topology a(Ro, L1 n L..) if and only if K is bounded in L1 + L.. norm, K is uniformly integrable, and K is uniformly Ro. (2.3.18) (Weak compactness in co.) Describe weak compactness in co by taking f = IN in the preceding. (2.3.19) (Conditional expectation counterexample.) Let (1k, F, p) be lR with Lebesgue measure, let f = 1i0,11, and let 9 = {S2, 0}. Then there is no

g E (L1 +L.)(fl,g,p) with

f

A

gdp=

JA

.

fdp

for allAEG. (2.3.20) (Alternative construction of the conditional expectation.) Let (Q, .F, p) be a a-finite measure space, and let G be a sub-Q-algebra of F.

Let B be the set of a-finiteness of p with respect to G. Then there is a probability measure P on .F and a Radon-Nikodym derivative p = dP/dp, where {p > 01 = B. We have

Eµ If 9] =

Ep [g/p 19] Ep [1/p 19]

is

This could be used as the definition of the conditional expectation in an infinite measure space. (2.3.21) (Uniformly integrable martingales.) Let (Sl,

P) be a probability space. Let (Ft)tEj be a stochastic basis indexed by a directed set J. (i) If (Xt) is a uniformly integrable martingale, then Xt converges in L1 to a random variable X and Xt = E-7t [X]. (ii) If (Xt) is a uniformly integrable submartingale, then Xt converges in L1 to a random variable X and Xt < E.Ft [X].

Proof. (i) Xt converges stochastically to a random variable X; and since it is uniformly integrable, it converges to X in L1 (1.1.3, 1.3.1, and 1.4.7). By Fatou's lemma, X E L1. Now ifs < t and A E F , then E [Xs 1A] = E [Xt 1A]

by the martingale property. Taking the limit as t -> oo, we get

E[X81A] =E[X1A] for all A E Fe. That is, X9 = Ewe [X].

80

Infinite measure and Orlicz spaces

(ii) Proofs for submartingales are similar.

(2.3.22) (Orlicz bounded martingales.) Suppose (Xt) is a uniformly bounded martingale. By the criterion of de La Vallee Poussin, there is an Orlicz function 4 with lim'(u)/u -+ oo such that E [4,(jXt1)] is bounded. (Modular boundedness.) On the other hand, if (Xt) is a martingale, and there is an Orlicz function 4' with 4'(u)/u -+ oo such that E [4'(IXtl)] is bounded, then (Xt) is uniformly integrable. What is less well known is that 4'( jXt I) is also uniformly integrable. To see this, observe that since 4'(IXtj) converges stochastically to 4'(JXJ), by Fatou's lemma we have E [,b(jXj)] < oo. But by Jensen's inequality,

4'(IXtI) = 4'(IEFt [X] 1)< E.rt [4'(IXI)]

The last term is uniformly integrable, so 4'(IXtl) is also uniformly integrable.

(2.3.23) (Orlicz norm and Orlicz modular convergence of martingales.) Let 4, be an Orlicz function with 4'(u)/u -+ oo. Let (Xt) be a martingale or a positive submartingale. (i) If (Xt) is bounded in D-modular: sup M.1 (Xt) < 00, t

then (Xt) converges in 4'-modular. (ii) Suppose also that 4, satisfies condition (02). If (Xt) is L4,-bounded, then (Xt) converges in L4, norm. Proof. The martingale case follows from the positive submartingale case by decomposition into positive and negative parts. So assume (Xt) is a positive

submartingale. The limit X of Xt exists in L1. Then t(Xt) converges stochastically to 4'(X), and by Fatou's lemma, 4'(X) E L1. By (2.3.21) we have

Xt < E.Ft [X], so by Jensen's inequality Eat [,P(X)].

As in (2.3.22), 4'(Xt) is uniformly integrable. Thus 4'(Xt) --> 4'(X) in L1. But 4' is convex, so

4'(x - y) «(x) - 4'(y) for x > y > 0. Therefore 'D (I Xt - X I) < 4'(Xt) -'D (X)

Thus 4'(IXt - XI) converges to zero in Ll; that is, Xt converges to X in 4'-modular.

2.3. Uniform integrability

81

(ii) If (A2) holds, modular convergence is equivalent to Lip convergence, and modular boundedness is equivalent to L4, boundedness (2.1.18).

Condition (02) cannot be omitted in (ii). See Mogyorodi [1978] and Bui [1987].

Remarks Conditional expectations in infinite measure spaces can be found in Chow [1960b] and Dellacherie & Meyer [1978].

3

Inequalities

In this chapter we will prove several different kinds of inequalities. We begin with the "three-function inequality," which relates weak inequalities, such as

tt{I9I>A}_a}

to strong inequalities, such as II91IP_ A} a}

If I dµ

or

bMM (b) 0. We claim that f, gn, hn satisfy the same hypothesis. Consider

A> 0. Ifti_1 ti},

1\1

{hn>A}={hn>ti}={h>ti}. Therefore

µ{gn > A} = µ{g > ti}
tj}

_

1

fdµ fdµ

ti

1 f A

fdµ.

This verifies that f, gn, hn also satisfy the hypothesis. Therefore the previous case yields

bM£(gb) 0, we have (S'g)

bMt (b) < aM4, (1) +aMt (b)

With an assumption, we get more:

(3.1.3) Theorem. Let 4) be an Orlicz function, and let

be defined as in (3.1.1a). Let f and g be measurable functions related by (Wg). If (Fg)

p{IgI > Al < oo

for all A > 0,

then for any a < b, (S9)

M

(g)

a

M

M. (f).

Proof. We may assume that f and g are nonnegative functions, since all quantities involved depend only on the absolute values of the functions. First, consider the special case where µ{g > 0} < oo (for example, µ(1) < oo) and g is bounded, say g < C. Then of course M M (b) J{g>o}

(b) dp < oo.

Thus we may solve (Sg) to get the required conclusion. Next, suppose only that g is bounded, but possibly µ{g > 0} = oo. For

e > 0, consider ge = (g - E)+. It, too, is bounded; and µ{ge > 01 _ i {g > E} < oo. Also, ge satisfies a weak inequality of the form (W9):

µ{ge>A}=µ{g>A+s}
a+e} fd< f

as required. By the previous case, we have

A

fdµ g .>a}

3.1. The three-function inequality

87

Now as e 1 0, we have gE T g, so by the monotone convergence theorem (and left-continuity of ) we get the conclusion (S.). Finally, consider general g. For a constant C > 0, let gC = g A C, so if A > C,

µ{gC>a}=0< 1 JgC>A} fdµ,

fdµ=1

p{gC>a}=µ{g>A}a}

f

gc>a}

fdµ.

Thus by the previous case, we have

M f

a

gC

Now as C T oo we have gC T g, so by the monotone convergence theorem we get (Se).

(3.1.4) Replace f by cf: If µ{IgI > Al Goo and

µ{I91>A}< c A

f

{IgI>a}

Ifldµ

for all A > 0, then for 0 < a < b, we have

MM (b) 0. The

inequality

µ{IfI >A}a}IfIdµ is clearly true for all A > 0, so by (3.1.3), if a < b, we have

M£(b)a}

IfIdµ

Suppose either (a)

µ{IfI>_a} 0; or (b) (D satisfies condition A2. Then (S9) holds for any a < b.

Proof. Clearly

µ{IfI>_A}< 1 A

111f I>a}

IfIdp,

so by the three-function inequality (3.1.2),

(Sf)

bMM(f) Al < oo for all A > 0. By Theorem (3.1.3), we have

(Sf)

MtIbI ucp(u),

so (u) = ucp(u) - D(u) < Cb(u). Thus MM (f /a) < CMS (f /a). Now if M£ (f /a) = oo, then also Mt (f /a) = oo, so (Sg) holds. On the other hand, if M£ (f /a) < oo, then Mg (f /b) < oo and we may solve (Sc) 'to obtain (Sf). The remainder of the proof is the same as the previous case.

(3.1.7) Replace g by g/c: If {If I > Al < oo and u{IgI ? ca}
a}

If I dp

3.1. The three-function inequality

89

for all A > 0, then for 0 < ac < b, we have

M

ac

(9) b

M

b-acM

a (')

(3.1.8) Take f = g: If {If I > Al < oo and p{I f I >_ A} 0, then for 0 < a < b, we have MgI

I

b

b

Note that we obtain the same conclusion (S9) in both (3.1.3) and (3.1.6). Which approach is better? In the typical application, we will have g > f, so that, assuming (F9), (3.1.3) is a stronger result than (3.1.6). For example, we will apply these results below where g is the maximal function constructed from f > 0 by 9 = suP Tn.f, n

where Tn are positive operators on function spaces, and To = I, so that f < g. Now if f < g, then (W9) is easier to verify than (W f). On the other hand, in infinite measure spaces, integrability of f implies (Ff) but not (F9). It is therefore of interest to know that in many cases, (W9) implies (Wf) with different constants. (See, for example, (8.2.5).)

The inequalities have all been stated using the nonstrict inequalities such as p{IgI > A}. They are equivalent to the same formulas with strict inequalities such as p{IgI > A}. This is because of the relations p{IgI > Al = limp{IgI > A + 1/n}, n

p{I9I > \} = limp{IgI > A - 1/n}. n

For example, by the monotone and dominated convergence theorems p{IgI > A}
a} IfI dµ

for all A > 0

is equivalent to

'

N-{I9I>AI < 1 f

{Ihj>A}

Ifldp

for allA>0.

In the case when 4 and the related function are Orlicz functions, our theorem tells us about Orlicz spaces, and Orlicz hearts.

Inequalities

90

(3.1.9) Corollary. Let 4 be an Orlicz function, and suppose that the function defined by (3.1.1a) is also an Orlicz function. Then: (a) L4. C LC; H4. C H. Suppose f and g are measurable functions with µ{IgI > Al < oo and

i4i

(W9)

lI

A}
0. Then: (b) f E L. r g E Lt.

(c) f EH4.

gEHH.

Proof. (a) If f E L4., then there is a > 0 so that Mp (f /a) < oo. Then by (2.1.19b), l;(u) _< -t(2u) so for b > 2a we have MM (f /b) < oo, and so f E Lt. Similarly, if f E H4., then for every a > 0 we have Mp (f /a) < oo. Thus for any b > 0, there is a with 0 < 2a < b, and we have M£ (f /b) < oo, so f EHH. For parts (b) and (c), proceed as in part (a), using (3.1.3).

(3.1.10)

Corollary. Suppose f, g are measurable functions such that

,u{I9l > A} < oo, and

i{I9I>A}< 1 A

f

If ldµ

{192.\}

for all A > 0.

(a) Let 1

b}

=b+bMPO (9b

b+b-aM'l (a Finally, set a = 1 and b = 2. The constants p/(p - 1) in part (a), (k + 1)/k in part (b), and 2 in the first inequality of part (c) are the best possible in this result. (See (3.1.14) and (3.1.15), below.)

Inequalities

92

Reverse inequality Theorem (3.1.2) has a companion "reverse" theorem. We take a = b and

f = h. If there is c so that

p{I9I>a}> c 1\

f

{If2a}

If ldµ

for all A > 0, then

M£(

)>cMi1

a I+cM£(al.

It is enough to prove it in the case a = 1, since we may replace A by aA. In fact, we will prove a more general result with two terms on the right. (It is not hard to see that it could be done with three or more terms on the right, but our applications use at most two terms.) Note that the constants c1, c2 are not required to be positive.

(3.1.11) Theorem. Let -t be a finite Orlicz function with 4D(u)/u --> oo, and let be defined by (3.1.1a). Let fl, f2, g be measurable functions. Suppose that there exist constants c1, c2 so that

Ap{Igl?A}>cl f fi >A} {If,ld+c2 I

f

f2ldp If2 I>A}

for all A > 0. Then we have Mt (9) >_ c1M4, (fl) + c1MM (fl) + c2M4, (f2) + c2M£ (f2)

Proof. This proof is similar to the proof of Theorem (3.1.2). (We take a = 1.) Again, we may assume that fl, f2, and g are nonnegative. Suppose first that fl, f2, and g are functions having countably many values, all in the set T = Cv U {0}, as before. Then

M£ (9) = f (g) dp =

f

`y (w (9)) du v(9)

z/i(y) dy dp o

apply F ubini's theorem to exchange the order

=f

J{g2:0(01

dµ0(y)dy

3.1. The three-function inequality

=

f

93

µ{g > V,(y)} b(y) dy

0

use the hypothesis > c1

0f00

f fi_(y)}

{fidpdy+c2

v(h) dy A dµ + c2

= ci f fo

f ff_

f0 f0

f2dµdy

0

w(f2)

dy f2 du

=clJ(fi) fi dz + c2 J O (12)f2du apply the case of equality in Young's inequality

f

= ci f 41 (fl) d i+ci + C2

f

IF ((p (fl)) dp

'D (f2) dp+c2

f

(f2)) dp

= ciM . (fi) + ciMt (fi) + c2M4, (f2) + c2Mg (f2)

For the case of general fl, f2i and g, approximate as follows. Given n, let countably many values 0 < < t_2 < t_1 < to be chosen so that

ti E Cv,, to > 2n, ti - ti_1 < 2-", limi__,. ti = 0, and ti/ti-1 < 1 + 2-n. Let

{gn = ti} = {ti < 9 < ti+1}, {gn = t0} = {to 91, {gn = 0} = {g = 0};

so that gn < g with gn -> g. Similarly, fln should be constructed to approximate f1 and f2n should be constructed to approximate f2. Then for ti_1 < A < ti we have {gn > Al = {g > ti}, and similarly for fl and f2. If c1 > 0, then c1

>

Cl

Cl

ti - ti-1(1 + 2-n) - A(1 + 2-n) If c1 < 0, then Cl

Cl

ti

A

Similarly for C2. Thus in the case where c1, c2 > 0, we have p{gn >_ A} = µ{g > ti} > -C1

f

ti {fl>}

f1 dj.t + tiC2 J

f2>ti}

f2 dp

Inequalities

94

fl. d1.1

c1

ti {h,.>a} Ci

A(l + 2-)

Mg (9n)

{fi,._a}

f2. dl.,

c2

+ C2 ti

{fzn>A}

fln dp +

c2

A(1 + 2-n) k2._!'\)

1 + 2-n M4, (fin) + 1 + 2-n

f2. dp.

M£ (fin)

C2

1+2 -n M (f2n) + 1+2 -n ME (fen C2

)

Take the limit as n -> oo to obtain the conclusion. If cl > 0 and c2 < 0, we have similarly Mg (9n) >_ 1 + 1

Mt (fln) + 1 + 2-n M£ (fln)

+ C2M4 (f2n) + C2M£ (f2n) .

Again, take the limit as n --> oo to obtain the conclusion. Cases with ci < 0 are similar. Theorem (3.1.11) has many simple consequences. The following will be included here. Others will be seen below.

(3.1.12) Corollary. Let 4) be a finite Orlicz function with 4)(u)/u --+ oo, and suppose that the function 1; defined by (3.1.1a) is also an Orlicz function. Suppose f and g are measurable functions satisfying

µ{IgI?A}> c

A

f

If Idti

{Ifl?a}

for all A > 0. Assume either (i) 4) satisfies (02) or (ii) i { If I > A} < oo for all A > 0. Then:

(a) g E Lt = f E L4,. (b) g E Ht

f E HiD.

Proof. We prove part (a). The other part is left to the reader. If g E Lt, then there is a > 0 such that Mt (g/a) < oo. But then we have Mt (f /a) < Mt (f /a) + Mt (f /a) < (1/c)Mt (g/a) < oo, so f E L.. Maximal inequalities for stopped processes The three-function inequalities may be used to prove refinements of the basic maximal inequality (1.1.7).

If (Xn) is an adapted process, let

Xiv = max IXnI, 1 A}, let the stopping time a be defined by o(w)

-{

inf{n:1 0, then (XN)NEIN is bounded in L£, so we have:

Theorem. Let 1 and 6 be as before. (i) If (XX)QEE is an integrable net bounded in Le, then X* E L. (ii) If (Xn) is a positive submartingale, bounded in Le, then X* E Lt. The estimates in (3.1.10) (3.1.13) apply.

Proof. For (ii), observe that if Xn is a positive submartingale, then M4, (X,) is an increasing function of a E E, since (by Jensen's inequality) P(Xn/a)

is a submartingale. Thus we have: If (Xn)nEIN is bounded in Le,, then (XX),EE is also bounded in Le, so X* E Lt. Complements

fo

(3.1.14) (Best constant.) The constant p/(p - 1) in Corollary (3.1.10(a)) cannot be improved. Let (D, .F, ,a) be (0, 1) with Lebesgue measure. Let

a satisfy -1/p < a < 0. Define g(t) = to' and f (t) _ (a + 1)ta. Then µ{IgI >_ Al = µ{ta > A} = µ{t < Al/«} = p((0, alga]) _ Al/" and

I f dy=(a+1)

fal/°

J

tadt

= a + 1 A(a+1)/a

a+1

= A z{IgI >_ A}.

But also

1

IIYIIp =

f tap dt < oo,

since ap > -1. Thus II9IIp/IIf IIp = 1/(a + 1). Finally sup I

1

-1

a+1

p

}---p-1.

(3.1.15) (Best constant.) The constant (k+1)/k in Corollary (3.1.10(b)) cannot be improved. Let (S1, F, p) be [0, oo) with Lebesgue measure. Let

Inequalities

96

a = -1/(k + 1), so -1 < a < 0. Then let g(t) = to and f (t) = (1 + a)ta. Check that

i{I9I?A}= 1 f

for allA>0.

If ldµ

{191>a}

Write 4k(u) = u(1og+u)k. The following are calculus exercises(!):

alga(-a)kk! M (a) _ (1 + a)l/a(a + 1)k+1'

bl/a(-a)k-1(k - 1)!

(gb)

M..k-

(a + 1)k

Then take a =IIf II'k and b = II9II4,k-,' and conclude [since a = -1/(k+1)] k

a _ (1 + a)a+1 b

So II9II

(-a)aka

k+1

'k-. = ((k + 1)/k) If II4,k

(3.1.16) (Improved constants.) Prove the elementary inequality:

alog+b ,(3A} < a f

X dµ.

{T> A}

Then, for 1 < p < oo, we have IiTllp < a/3P

(p P 1)

IIXIIPI

To see this, apply the three-function inequality (3.1.2) with: -t(u) = up/p,

cp(u) = 0-1, (u) = uP/q, where q = p/(p - 1) is the conjugate index.

3.1. The three-function inequality

97

Then with f = aX, g = T//3, h = T, the hypothesis of Theorem (3.1.2) is satisfied. The conclusion is 1

gbP_1

P a) aP - b IITII' - pa-1

(1

IIXIIP

Substitute a = 1 and b = q/3P to obtain the result. (3.1.18) (Best constant.) Show that aoPq is the best constant in the preceding. It is clearly enough to consider the case a = 1. For ,3 < 1, use the measure space (0, 1], and T(w) = w8)(3-118, X(w) _ (s + 1)w8, where -1/p < s < 0; then let s -+ -1/p to show that the constant is best possible. What should be done for ,3 > 1 ? (3.1.19) (Hardy-Littlewood.) The best known use of maximal functions

is due to Hardy and Littlewood. Let SZ = (0,1) and let It be Lebesgue measure. Suppose f : (0, 1) -+ IR, is an integrable function. Define

f*(x)=sup

fv

vu uJ 1

If(t)Idt:0 Al < J

f (t) dt. g>A}

This yields, then, many corollaries. If 1 < p < oo, and f E Lp, then f* E Lp, and IIf*IIP 0. Let o be the stopping time equal to the first n such that Xn > A. The set for = k} E.Fk, hence n

E [1{1 1. Then n

Zn=X1-J:Yi=c-(Xn-c)=2c-Xn. i=2

Thus supn Zn = 2c - inf Xn = 2c - 0 = 2c. In an infinite game, the gambler can double his initial fortune. Even if an infinite game is impractical, it remains that in a game of sufficient duration, a gambler endowed with very large reserves or credit can double his initial fortune with large probability. The following maximal inequality, due to Burkholder [1986], allows for V with both signs. In particular, it gives the bound 2 in the case of the double-or-nothing martingale. The case a = 0, b = 1 is Theorem (3.2.1).

(3.2.2) Theorem. Let a, b be real numbers with a < 0 < b. Let X be a martingale, let V be a predictable process with values in the interval [a, b],

andletZ=V*X. Then forallA>0, P{Z* > Al
(V - a)Y,

(3.2.2a)

=1

n

Bn = E(b - V )Y.

(3.2.2b)

i_1

Then (3.2.2c)

An + Bn = (b - a)Xn,

(3.2.2d)

bAn + aBn = (b - a) Zn, and

(3.2.2e)

(An - An_1)(Bn - Bn_1) > 0

(with the convention that AO = Bo = 0). Let u : JR2 -+ 1R, be defined by

u(x , y)

1 + xy

if IxI < l and jyj < 1

Ix + yl

otherwise

Note that u is continuous, since the two formulas agree on the boundary square. Next, 0 < u(x, y) - I x+yi 1 }. Now the stopped process ZTnn is the transform by V of the stopped martingale Xrnn. Thus we have from (3.2.2j)

P{IZrnnl > 1} < (b - a)IIXTnfhi1 But IXni is a submartingale, so by (1.4.18), we have IIXTnnII1 1} < (b - a) IIXII1.

Applying this to the martingale X/A yields the maximal inequality stated. In the proof that the constant b - a is best possible, we may assume that A = 1. If a = 0, we may assume b = 1. Then the deterministic example Xn = Vn = 1 shows that the constant 1 is the best possible (also in Doob's inequality). If a < 0, consider a martingale X such that X1 is the constant 1/(b - a) and X2 = X3 = is such that P JX2 =

2

a

-a 2(b - a)'

P{X2=0}= 2b-a

2(b - a)*

Let V1 = b and Vn = a for n > 2. Then P { Z * = 11 = 1 and I I X I I 1 = 1/(b - a). In Section 6.2, below, we give the proof that the transform of an L1bounded martingale converges a.s. Remarks

References related to this section are: Burkholder [1966] and Burkholder [1986].

3.3. Prophet compared to gambler Let n E IN, and let the random variable Xi be the fortune of a player at time i, for 1 < i < n. We may sometimes use a boldface letter as an abbreviation: X = (X1, X2,- .. , Xn). The player is to stop playing at a certain time i; his goal is to choose i so that his fortune Xi is as large

Inequalities

104

as possible (on the average). We will consider two types of players: the prophet and the gambler. A prophet is a player with complete foresight. Since he knows all the values Xi, he may simply choose the largest one, and stop at the appropriate time. Thus he achieves the maximum X1 V X2 V V X,,,.

A gambler is a player without knowledge of the future; but he knows the past and the present, and has knowledge of the odds (that is, the joint distribution of X). Let Fi be the v-algebra generated by X1, X2, , Xi. Then the time the gambler chooses to stop must be a stopping time for this stochastic basis (-Fi) 1. Write E,,, for this set of stopping times; their values are in the set {1, 2,. . . , n}. When the gambler uses the stopping time z E E,,,, his fortune when he stops is XT. So his expected fortune is E [XT]. Thus the best expected fortune he can achieve is:

sup{E[XT] : r E En}. This is known as the value of the process X = (X1, X2,

V = V(X) =

,

X,,,). Notation:

V(X1,X2,... ,Xn).

We will be interested in prophet inequalities. They compare the expected

gain P of a prophet to the expected gain G = V of a gambler. Of course the prophet has the advantage, so P > G. A surprising result is that often there are moderate universal constants C (independent of n and of the distributions of the Xi) such that P < CG. In the independent positive

case, C = 2 (Theorem 3.3.2) and the constant 2 is optimal. Thus the advantage of knowing the future is not as large as might have been expected.

To be sure, the prophet uses his foresight only to stop the game, not to change the stakes. The gambler uses his nonanticipating skills for the same purpose. A different situation arises when the players are allowed to change

the stakes. A prophet inequality (with optimal constant 3) also exists in that case. This will be treated in Theorems (3.3.4) and (3.3.5). Stopped processes

We prove first a basic lemma; the technique used to define the stopping times of is known as "backward induction," since we proceed from Qi+i to Oi.

(3.3.1) Lemma. Let X1, X2,. , Xn be independent nonnegative random variables. Define an = n, and inductively for i = n - 1, n - 2,... , 2, 1: (3.3.1a)

Qi =

i

on {Xi>E[Xoi+1]}

ai+1

on {Xi < E [Xoi}1] }.

Then we have n-1

E [(Xi - E

E [Xn] + i=1

[Xoi+1+]

= E [Xoi]

3.3. Prophet compared to gambler

105

Proof. Let i be given, 1 < i < n. Then we have (3.3.1b)

(Xi

- E [Xi+J)+

= 1 {xi<E[xoi+i]} (Xi - E [Xoi+1])

Since 1{Xi <E[x,. ] } and XQi+, - E [x,,+,] are independent, we obtain

E [1{xi<E[xai+i]}

=E

(Xvi - E [Xoi+i])]

[1 {xi <E[xoi+1 ] }

(Xoi+i - E [Xai+i ]) ]

(3.3.1c)

= P {Xi < E [Xoi+i ] } E [Xoi+, - E [XQi+1 ] ] = 0.

Integrating (3.3.1b) and adding (3.3.1c) to it, we obtain

E [(Xi - E

(3.3.1d)

[Xoi+1])+]

= E [Xoi] - E [Xi+,]

Summing these equations from i = 1 to i = n - 1 and adding E [Xn] _ E [Xan], the assertion follows.

Now assume n > 2. Write V = V(X1i X2, , Xn), ei = E [Xi], and 1V = V (X2i , Xn). Clearly 1V < V, so the theorem below proves slightly more than P < 2V. (3.3.2) Theorem. Let X1, X2, , Xn be independent positive random variables with 0 < E [Xi] < oo. Then

E[(X1VX2V...VX, -1V)+] 0. Then p(Xo) < 0 while p(X,,) = E [Xn] (1/2)E [IX. - E [Xn] 1] > E [Xn] - (1/2)E [IXnI] - (1/2)E [Xn] > 0. Here we will not require the independence of the X2's; only that the centered random variables Xi - E [Xi] = Xi - e2 be differences of a weak martingale. A sequence Y2 is a weak martingale if E [Y2 I Y2_1] = Yi_1 for all i. This notion is of interest in convergence theory, since Ll-bounded weak martingales converge in probability, while the issue of a.e. convergence remains open (Nelson [1970]). Before we prove the theorem, we insert two lemmas.

(3.3.3) Lemma. Let X be an integrable random variable, and let a < b be constants. Then

(b - a)P{X > b} < E [(X - a)+] - E [(X - b)+] < (b - a)P{X > a}. Proof. Note that

E [(X - a)+] - E [(X - b)+] = E [(X - a - X + b)1 {x>b}] + E [(X - a)1{a<x e, then by the second inequality in (3.3.3),

E [(X - e)+] < E [(X - d)+] + d - e. Next, if d < e, then by the first half of (3.3.3),

E [(X - e)+] < E [(X - d)+] + (d - e)P{X > e} = E [(X - d)+] + (d - e) + (d - e)P{X < e} < E [(X - d)+] (1 + P{X < e}) + d - e because E [(X - d)+] > E [X - d] = e - d.

(3.3.5) Theorem. Let X = (Xo, X1,

, X,) be an integrable stochastic process. Let 9i be a-algebras such that Xi is Suppose that gi-measurable.

(3.3.5a)

E [Xi I Xi_1] = ei

is constant. Assume p(Xo) < p(X.) (or assume that all the Xi's are nonnegative). Then the expected gain P of the prophet and the expected gain G of the gambler are related by:

P ei, and E [(Xi - e2)+] < E [(ei+1 - ei)+] = ei+1- ei, with strict inequality unless ei+1 = e2 and Xi = ei. But by equality in (3.3.5c), we must have equality in every term E [(Xi - ei)+] < e2+1 - ei, so Xi = ei =

foralli. There is a version depending on n, obtained by applying the stopping time result to transforms. Note that if the random variables X1i , Xn are nonnegative, then we can always add X0 = 0 to the beginning without changing P or G.

Inequalities

110

(3.3.6) Proposition. Let X = (Xo, X1i

, Xn) be nonnegative random variables such that E [Xi I Xi+l] = ei is constant, and Xo = 0. Then

n

P en (both conditions hold if Xn = eo). Then

P3 ei}. Hence n

(E [(ei - Xi-1)+] + E [(ei - Xi-1)

G3 =

])

i=1

On the other hand, the difference of the summands is

E [(ei - Xi-1)+]

- E [(ei - Xi-1)

] = ei - ei-1,

hence

n

n

E [(ei - Xi-1)+] -

G3 = 2 i=1

E [(ei - ei_1)-] = 2G - en + eo. i=1

3.3. Prophet compared to gambler

111

Similarly, P. = 2P - en + eo.

Now P < 3G and eo > en imply P8 < 3G8. If en = eo, then P. = 2P and G8 = 2G. Complements

(3.3.8) (Optimality of backward induction.) In Lemma (3.3.1), we have E [X 1] = V and E [X2] = 1V. It follows inductively (backward on i) that E [X,,] is the value of the process (Xi, , Xn). (See Chow, Robbins & Siegmund [1971], p. 50.)

(3.3.9) (Constant 2 is best in (3.3.2).) Let n = 2, X1 = 1 and X2 = M > 1 with probability 1/M and X2 = 0 with probability 1-1/M. The expected fortune of the prophet is

=2-M. The gambler expects to receive 1 regardless of when he stops. Then P/V = 2 - 1/M, so 2 is the best constant since M is arbitrary. (3.3.10) (Bounded case.) In Theorem (3.3.2), if the random variables take

values in [0, 1], then the prophet inequality P < 2V can be improved to P < 2V - V2 (Hill [1983]). (3.3.11) (Constant 3 is best in (3.3.4).) Given e > 0, there exist random variables Xi, X2, ... , X,, with P' > (3 - 6) G', where P and G' are the expected gains of the prophet and of the gambler for the primed process. The sequence (Xi') is obtained from a sequence (Xi) by inserting many copies of the constant random variable E [Xi] between Xi and Xi+1. The independent positive sequence Xi itself is chosen so that En, E [(X2 - ei)+] > (2 - e)G and Xo = Xn (Krengel & Sucheston [1987], p. 1598). (3.3.12) (An economic interpretation of (3.3.4).) Assume that (Yn) and (Zn) are two integrable processes with arbitrary distributions, but independent of each other. Players observe the two processes alternately: say

Xn = Yn for n even and Xn = Zn for n odd. Then the adjacent Xn's are independent. To fix the idea, suppose that a conglomerate is sufficiently diversified so that the stock prices of two of its firms, F1 and F2, are independent. Let Yn be the value of cl shares of Fl; let Zn be the value of c2 shares of F2. Two of the conglomerate's executives are allowed to trade each year kc1 shares of F1 for kc2 shares of F2 or vice versa. Each executive chooses each year his own value of k, but the k's are bounded by a fixed constant C. Assume that the junior executive bases his decisions on the present (equivalently, present

and past): he is a gambler. The senior executive knows the future: he is a prophet. In practice, there is a device equivalent to the gift of prophecy put at the disposal of senior executives: the right to exercise options, i.e., trading in stock some time in the future at prices prevailing earlier. The theorem implies that the expected gain of the senior executive is less than 3 times that of his junior colleague (Krengel & Sucheston, unpublished, 1987).

112

Inequalities

Remarks

Prophet inequalities were introduced by Krengel & Sucheston [1978], where it

was first shown that there are constants C such that P/V < C. For stopped processes, Krengel and Sucheston obtained C = 4; the same paper contains a proof by D. Garling that C = 2 can be taken. The paper also studies the analogous but more difficult problem when the X2's are averages of independent random variables, obtaining C = 5.46. The problem whether the best constant in this case was 2 had remained open until Hill [1986] resolved it in the affirmative. The proof of Theorem (3.3.2) given here is based on arguments communicated to us by D. Gilat and R. Wittmann. Theorems (3.3.4) and (3.3.5) are from Krengel & Sucheston [1987]. Proposition (3.3.6) was observed by Sucheston & Yan [in press].

4

Directed index set

In this Chapter we present the theory of martingales and amarts indexed by directed sets. After Dieudonne showed that martingales indexed by directed sets in general need not converge essentially, Krickeberg-in a series of papers-proved essential convergence under covering conditions called "Vitali conditions." This theory is presented in an expository article by Krickeberg & Pauc [1963] and in a book by Hayes & Pauc [1970].

Here we offer a new approach and describe the subsequent progress. The condition (V), introduced by Krickeberg to prove the essential convergence of Ll-bounded martingales, was shown not to be necessary. Similarly the condition (V°), introduced to prove convergence of Ll-bounded submartingales, is now also known not to be necessary. The condition (V.), which Krickeberg showed to be sufficient for the convergence of martingales

bounded in the Orlicz space L,,, is also necessary for this purpose if the Orlicz function 1 satisfies the (02) condition. In each instance, the convergence of appropriate classes of amarts exactly characterizes the corresponding Vitali condition. This is of particular interest for (V) and (V°) since there is no corresponding characterization in the classical theory. In general, to nearly every Vitali type of covering condition there corresponds the convergence of an appropriate class of "amarts." The understanding of this fact was helped by new formulations of Vitali conditions in terms of stopping times. Informally, a Vitali condition says that the essential upper limit of a 0-1 valued process (1A,) can be approximated by the process stopped by appropriate stopping times. This has a clear intuitive meaning even in the case of multivalued stopping times because, as a condition for convergence, the overlap of the values is small, in a precise sense. The application of martingale theory to derivation theory has been long known, but also amarts come into their own. A derivative of a superadditive set-function is both a supermartingale and an amart (4.2.18). In the classical setting of the derivation theory the Vitali condition (V) holds (4.2.8), but (V°) does not. So supermartingales need not converge essentially, and the amart theory is needed to prove convergence. Similarly, derivatives of functions of measures are amarts, and the Riesz decomposition sheds some light on their behavior (4.2.19). Since the condition (V) is not necessary for essential convergence of martingales, is there a covering condition both sufficient and necessary? The

answer is yes: There is a condition (C) for this purpose. Condition (C)

Directed index set

114

is sufficient for convergence of Ll-bounded martingales, and also necessary

if the index set has a countable cofinal subset. The question of necessity remains open for the general index set. It is also not known whether (V4.) is necessary for convergence of Lw-bounded martingales if (02) falls. Another open question is the existence of a covering condition both sufficient and necessary for convergence of Ll-bounded submartingales. The conditions (V4.) are modeled on similar covering conditions in the theory of derivations (see Chapter 7). Condition (C) has moved in the opposite direction-it was first introduced in the study of convergence of processes indexed by directed sets, then translated into a condition for derivation theory. As with Doob's theory of martingales indexed by IN, the main interest and the main difficulties occur in the L1-bounded case. The greater part of the Chapter is devoted to that case.

4.1. Essential and stochastic convergence There are two modes of convergence that will be used in this chapter: essential convergence and stochastic convergence.

When the index set is countable, essential convergence coincides with almost everywhere convergence, but when the index set is uncountable, essential convergence is still reasonable, although almost everywhere convergence may not be. Essential convergence is called order convergence in some of the literature. Stochastic convergence is also known as convergence in probability (or in measure). We will consider the corresponding stochastic upper limit and stochastic lower limit. Essential convergence

Let (St, F, P) be a probability space. Let S be the set of all extendedreal-valued random variables. Let P C S be some set of random variables. The random variable Z E S is called the essential supremum of the set P if

(1) Z > X a.s. for all X E P; (2) if Y E S and Y > X a.s. for all X E P, then Y > Z a.s. (Thus Z is the least upper bound of P in the partial order obtained by identifying functions that agree almost everywhere.)

(4.1.1) Proposition. (a) Every subset P of S has an essential supremum, ess sup P, unique up to null sets. (b) There exists a sequence Xn in P such that sup,, Xn, = ess sup P a.s. (c) If the family P is directed, the sequence X,,, may be chosen to be a.s. increasing. Proof. Assume first that all X E P satisfy 0 < X < 1. Let Pl be the set of all countable suprema supk Xk for Xk E P. Write

a=sup{E[Y]:YEP1}.

4.1. Essential and stochastic convergence

115

(Since 0 < Y < 1 for all Y E P1, we have 0 < a < 1.) For each n, choose Yn E P1 with E [Yn] > a - 2-n, and define Z(w) = supra Y,,(w). Then Z is itself a countable supremum of elements of P, so Z E P1. We have E [Z] =a. We claim that Z is an essential supremum for the set P. If X E P, then

XVZEP1and

a>E[XVZ]>E[Z]=a,

so X V Z = Z a.s., hence X < Z a.s. On the other hand, if Y is a random

variable with Y > X a.s. for all X E P, then Y > Z a.s., since Z is a countable supremum of elements of P. For the general case, choose a continuous strictly increasing bijection between [-oo, oo] and [0,1], and apply the preceding case. For the uniqueness, suppose that both Z1 and Z2 are essential suprema

for the set P. Now Z1 > X a.s. for all X E P, so (since Z2 is essential supremum), Zl > Z2 a.s. Similarly, Z2 > Z1 a.s. Therefore Z1 = Z2 a.s. This completes the proof of (a) and (b). For (c), note that if the sequence Xn is chosen for (b), we may choose Xn E P so that X;, > Xn and X,, > Xk for 1 < k < n since P is directed. Then Xn is increasing and sup Xn = ess sup P.

We will write ess sup P for the essential supremum of the set P. The essential infimum is defined analogously (the greatest lower bound in the partial order obtained by identifying functions that agree almost everywhere), or equivalently in terms of the essential supremum: ess inf P =

-(ess sup{-X : X E P}). Note: it is easily verified that if P is countable, then Z = ess sup P is the pointwise supremum,

Z(w)=sup{X(w):XEP}. This need not be true for uncountable P. For example, if (fl, Y, P) is [0, 1] with Lebesgue measure, and P is the set

P={1{X}:xE[0,1]}, then ess sup P is 0 a.s., but sup { X (w) : X E P } is identically 1. If P is replaced by a nonmeasurable subset, then sup { X (w) : X E P } is not measurable. The essential supremum is better behaved.

(4.1.2) Proposition. Suppose the set P is a nonempty family of indicator functions of measurable sets. Then ess sup P is also an indicator of a measurable set.

Proof. Let Z = ess sup P. Since P is nonempty, there is X E P, and thus Z >_ X > O a.s. For all X E P, we have X < 1, so Z < 1. Let A = {Z > 1}. We claim Z = 1A a.s. If X E P, then X < Z a.s. But X is an indicator function, so it is 0 whenever it is less than 1, and we have

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116

X < 1A a.s. Therefore, by the definition of essential supremum, Z < 1A. Clearly Z > 1A a.s., so Z = 1A a.s. If C C_ F is a family of measurable sets, then we will write B = ess sup C for the set of the proposition; that is,

1B = esssup 1A. AEC

(By convention, if C = 0, then ess sup C = 0.)

(4.1.3) Definition. Let J be a directed set, and let (Xt)tEJ be a net of random variables. The essential upper limit of (Xt) is defined by e lim sup Xt = ess inf ess sup Xt. 8EJ tEJ tEJ t>s

We will often write X * for the e lim sup of a process (Xt). The essential lower limit e lim inf Xt is defined analogously: Xt = ess sup ess Einf Xt, sEJ t>8

e

or e lim inf Xt = - e lim sup (-Xt). (4.1.4) Proposition. If (Xt) is a net of random variables, then we have e lim inf Xt < e lim sup Xt a.s.

Proof. For each s E J, ess inft>8 Xt < X8. Therefore, for each u E J, ess sup ess inf Xt < ess sup X8, s>u

t>s

s>u

hence e lim inft Xt < ess sup.>u X8. Therefore, e lim inf Xt < ess inf ess sup X8 = e lim sup X8. t

u

s>u

s

Let (Xt) be a net of random variables. It is said to converge essentially to a random variable X,,. if X00 = e lim sup Xt = e lim inf Xt tEJ tEJ

a.s.

Then we write X,,. = e limtEJ Xt. We leave to the reader the verification that when J is countable, we have (e lim sup Xt) (w) = lim sup (Xt (w)), tEJ tEJ (e lim inf Xt) (w) = lim inf (Xt (w) ), tEJ tEJ

(elimXt)(w) = lim (Xt(w)) tEJ

for almost all w.

tEJ

4.1. Essential and stochastic convergence

117

Stochastic convergence

Stochastic convergence (convergence in probability, convergence in measure) will be useful not only in situations when essential convergence fails,

but also as a tool in proofs of essential convergence. We begin with the stochastic upper and lower limit. (There does not seem to be a reasonable notion of "stochastic supremum," however.) Let (Sl, P) be a probability space.

(4.1.5) Definition. Let (Xt)tEJ be a net of extended-real-valued random variables. A random variable Y such that limt P{Xt > Y} = 0 is called asymptotically greater than Xt in probability. (The words "in probability"

will sometimes be omitted.) The stochastic upper limit of (Xt), written slim sups Xt, is the essential infimum of the set of all extended random variables which are asymptotically greater than Xt in probability. The stochastic lower limit, written slim inf Xt, is slim inf Xt = - slim sup (-Xt). If slim sup Xt = slim inf Xt = X,,,,, then X,,. is called the stochastic limit of (Xt), which is then said to converge stochastically (or to converge in probability) to X,,,,. We write slimtEJ Xt = X11There are relations between stochastic convergence and essential conver-

X.

gence.

(4.1.6) Proposition. Let (Xt)tEJ be a net of extended-real-valued random variables. Then (4.1.6a)

e lim inf Xt < slim inf Xt < slim sup Xt < e lim sup Xt.

Therefore, if (Xt) converges essentially, then it converges stochastically to the same limit.

Proof. If so E J, then esssupt>80 Xt > X8 for all s > so. Hence lim P ( ess sup Xt < X3 I

l

0.

t>80

ess inf ess sup Xt > s lim sup X3; 80

t>80

8

this holds for all so, so e lim sup Xt > slim sup Xt. tEJ tEJ Similarly,

e lim inf Xt < s lim inf Xt. tEJ tEJ

M

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118

(4.1.7) There is a natural definition intermediate between essential convergence and stochastic convergence. When three of the four random variables in (4.1.7a) coincide, we say that (Xt) demiconverges to their common value. More specifically, we may say that (Xt) upper demiconverges to Xoc if slim Xt = e lim sup Xt = X,,,, t

t

and that (Xt) lower demiconverges to X,,. if

slimXt = eliminf Xt = Xoo. Thus demiconvergence implies stochastic convergence. (For some results on demiconvergence, see (7.3.24), (9.1.2), (9.4.4), (9.4.13).) Let (At) be a net of events. Then slim sup lAt is an indicator function. Indeed, if Y is asymptotically greater than (1At), then

0=limP{1At >Y}>P{O>Y}, so Y > 0 a.s. Thus {1At > Y} > {1At > 1{Y>1}}; that is, 1{Y>1} is also asymptotically greater than (1At), and 1{Y>1} 1c I, hence limt P(At \ C) = 0 if and only if 1c is asymptotically greater than 1At in probability. It can be easily verified that if X,,. is finite a.s., then slimtEJ Xt = Xoo is equivalent to the usual definition: (4.1.7b)

for alle>0.

lim P { j x"" - Xt I > 61 = 0 tEJ

4.1. Essential and stochastic convergence

119

For sets, B = slim At holds if and only if (4.1.7c) P(At L B) -> 0. (4.1.8) Lemma. Suppose (At) is an adapted sequence of sets. Then A = slim sup At is the largest set A such that (4.1.8a)

limsupP(At n B) > 0 t

for all subsets B C A of positive probability. Proof. Let A be a set such that (4.1.8a) holds for every B C_ A of positive probability. Then

P(Atn(A\A))

P(At\A)-*0,

which implies P(A \ A) = 0 and therefore A C_ A a.s.

Conversely, let B C A with P(B) > 0 but suppose that (4.1.8a) fails. We have

P (At \ (A\ B)) < P(At \ A) + P(At n B).

Applying (4.1.7a) with C = A, we obtain P(At \ A) = 0. Therefore limP(At \ (A \ B)) = 0, hence P(B) = 0, because A is the smallest set C as in (4.1.7a).

(4.1.9) Lemma. Let A = slim sup At, and let (s,,,) be a sequence of indices. Then there exists an increasing sequence (t,,) of indices such that sn < to and A C U001 At . Proof. On measurable subsets of A define a function y by y(B) = lim sup P(At n B). t

Set B1 = A and choose t1 > s1 such that P(At1 nB1) > y(Bi)/2. Then set B2 = A\At1 and choose t2 > t1, t2 > s2, such that P(At2 nB2) > y(B2)/2. Continue the definition of (tn) and (Bn) recursively: given t 1 , , to and 113n, set B1,--_ n

Bn+1=A\UAt, i=1

and choose to+1 > tn, to+1 >_ Sn+l, with P(Ati}1 n Bn+1) >_ y(Bn+1)/2. Since

00

00

nBn=A\UAt,, n=1

n=1

it suffices to show P(fn Bn) = 0. The sets B. n At. are pairwise disjoint in a finite measure space, and y(Bn) decreases, hence

limy(Bn) = limlimsupP(At n Bn) < 2limsupP(Atn n Bn) = 0. n

It follows that y(n Bn) = 0, hence by Lemma (4.1.8), P(l Bn) = 0, so A C Un At,, .

120

Directed index set

Proofs of convergence in Chapter 1 (for example (1.2.4)) are based on

approximation of the a.s. lim sup by the stopped process. We will see below that also on directed sets the stochastic upper limit slim sup At can be approximated by a stopped set A(r), with r E E. This was implicitly used in the proof of (1.3.9). (For the essential lim sup, a lim sup At, such an approximation need not hold; it must be postulated; this is the meaning of the Vitali conditions below.) In fact, it suffices to stop with ordered stopping times. A simple ordered stopping time is a stopping time r : SZ -+ J such that its set of values is totally ordered (see Section 1.3). We write E° for the set of ordered stopping times.

(4.1.10) Proposition. Let (At) be an adapted family of sets, and write A = slim sup At. For every e > 0 and every to E J, there is a T E E° such that T > to and P(A A(T)) < e.

Proof. Given e and to, choose a sequence sn E J such that s,,, > to and P(At \ A) < e2-n-1 for all t > sn. Let (tn) be the sequence obtained by application of Lemma (4.1.9). Choose k such that P (A \ Uki Ai) < e/2. Define r = to on At. \ U 1 At; for n < k and r = tk+1 on the rest of Q.

Then rEE0and P(ADA(r))<e. The corresponding result for processes is proved next. A process (Xt)tEJ is called asymptotically uniformly absolutely continuous if, for every e > 0,

there exists s E J and 6 > 0 such that if P(B) < 6 and t _> s, then E [IXt 1B1] < e.

(4.1.11) Stochastic maximal inequality. Let (Xt)tEJ be a positive adapted process, let A > 0, and define A = Is lim sup Xt > A}. (a) Then

P(A)
0. Fix a number a with

0 A} C s1imsupBt = B. Indeed, suppose Y is a random variable asymptotically greater than (Xt), and D is an event such that 1D is asymptotically greater than (1Bt). If

Yt_ (A - a t

Y

onQ\D on D,

4.1. Essential and stochastic convergence

121

then we claim Y' is asymptotically greater than (Xt):

P{Xt > Y'} < P{Xt > Y} + P{Bt \ D} -. 0. Therefore slim sup Xt < Y. Thus {slim sup Xt > Al C {Y' > A} C D.

This is true for all events D such that 1D is asymptotically greater than

(IBj, so {slim sup Xt > Al C B.

Now given e > 0, choose s E J and T E E° such that T > s and P(B 0 B(T)) < e. This is possible by (4.1.10). Then we have

P(B) < P(B(T)) + e
0, choose s E J and e > 0 with e < 6 such that P(B) < 2e and r > s implies E [X.r 1B] < 6A. Then let a < A/2 be so small that (4.1.11a)

P (s lim sup{Xt > A - a} \ A) < E.

Write Bt = {Xt > A - a} and b = slim sup Bt. Now choose T E E°, T > s, such that P(BOB(T)) < E. We have A C B, so P(A\B(T)) < P(B\B(T)), so by (4.1.11a) we have E [XT 1B(,r)] < E [X.r 1A] + 6A.

Then:

P(A) < P(B) < P(B(T)) + e < 1 E[XT1A]+ 6A

A-a
0 and 6 -a 0.

In the previous result, note that if (Xt) is a positive submartingale or positive supermartingale, then the right hand side of the inequality in (a) simplifies to (1/A) limtE j E [Xt].

We will now use the stochastic maximal inequality to show that the (essential) maximal inequality is equivalent with convergence of martingales. We state it here in a somewhat abstract form in part (ii); the best known application is M(A, e) = e/A, as in part (i).

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122

(4.1.12) Proposition. Let (.Ft)tEj be a stochastic basis. The following are equivalent:

(i) For every integrable real random variable X, for every A > 0, we have

P{elimsupIEFt[X]I >A} < E[IXI]. (ii) There exists M : IR+ x IR+ -.' 1R.+ with (a) lim6_.,o M(A, E) = 0 for

every A > 0, and (b) for e, A > 0 and every positive integrable random variable X with E [X] < E, we have

P{elimsupEFt [X] > A} < M(A,e).

(iii) For every positive integrable random variable X, the martingale Xt = E.*Ft [X] converges essentially.

(iv) For every integrable real random variable X, for every A > 0, letting A = {elimsup IEFt [X] I > Al, we have

P(A) < Proof. (i)

E [IXI 1A] .

(ii). Set M(A, e) = e/A.

(ii) = (iii). Suppose the function M as in (ii) exists. Let X E Ll. Write J:=, = v(U.Ft). We claim that the martingale Xt = E.171 [X] converges essentially to EF°° [X]. Since X may be replaced by EF°° [X], we may assume that X is .F,,.-measurable. In that case, we must show that

elimXt = X.

Fix a > 0 and A > 0. Choose e > 0 so small that e < aA/2 and M(A/2, e) < a. Since X is F,,.-measurable, there must exist to E J and Y E L1(.7=to) such that E [IX -Yj] < E. (See (1.1.12).) Then by the triangle inequality, for t > to,

P {e lim sup jXt - X > Al < P {e lim sup E.97t [IX - Yj] > 2 }

+P{elimsup E-7-[IX-Yl]> 2}. But the first term on the right hand side is at most M(A/2, e) < a, while by Chebyshev's inequality the second term is at most 2e/A < a. Therefore P {e lim sup l Xt - X1 > A} < 2a for all t _> to. Thus a lim sup jXt - X = 0, and therefore elimXt = X. (iii) (iv). Let X E Ll. Let Xt = E.1t [X], so that also X,. = EF, [X] for ,r E E by the localization theorem (1.4.2). Since Xt converges essentially,

4.1. Essential and stochastic convergence

123

we haves lim sup Xt = e lim Xt = e lim sup Xt and A = {slimsuplXtI > A}.

Now if Yt = IXtj, then (Y,),EEo is uniformly absolutely continuous by (2.3.12). Then applying (4.1.11(b)), we obtain

P(A) < 1 limsupE [Y, 1A] A rEE° < 1 limsupE A rEE°

[IXI] 1A]

_ E [IX1 1A] . The last equality follows because Ear [IXI] converges to E-"00 [IXI] in Ll. (iv) (i) is obvious The preceding proof in fact shows more. We need not consider all random variables X, but only a certain subclass. A family .F of processes (Xt, fit),

where Xt is measurable with respect to Ft, is called stable if for every (Xt, Ft) E £ and every to E J, the process (Yt, Gt) is also in £, where, for t > to, Yt = Xt - Xt,, and Gt = Ft; and for other t, Yt = 0 and Gt = Ft.. When we say (Yt) E £, without specifying the v-algebras, we understand that (Yt, Gt) E £, where Gt is the a-algebra generated by { Ys : s < t }.

(4.1.13) Proposition. Let (.Ft)tEJ be a stochastic basis, and let £ be a stable subfamily of the family of uniformly integrable martingales. Then the following conditions are equivalent:

(i) For every (Xt) E £ and every A > 0, we have

P{elimsupIXtj >A}
t

(4.1.14) Proposition. Let (Zt) be the Snell envelope of the integrable process (Xt). Then: (1) If a E E, then Za = ess sup

[Xr]

.

TEE r>a

(2) For or E E, there exists a sequence T,+, E E, T,,, > or, such that [XT..] I Z,.

(3) (Zt) is an amart and a supermartingale.

Proof. (1) Let or E E, and let t be one of the values of or. If T E E and

T > a, then there is r' E E, r' > t, such that r = T' on for = t}. Indeed, choose to with to > r; and let r ,r(w) to

if a(w) = r(w) otherwise.

Similarly, if T' E E and T' > t, then there is r E E, r > o, such that r = T' on for = t}. So on the set {a = t}, we have (by the localization theorem (1.4.2))

E*F° [Xr] = E.rt [Xr] =

[XT,]

.

Hence on {a = t} we have ess sup E'F° [Xr] = ess sup Et [XT] = Zt = Za. r>a r>t

This is true for all t. (2) First consider the case or = t constant. For Tl,T2 > t, if we define B = {Eat [XT.]

Eat [Xr2] },

then B E Ft, and r defined by Tl T =

{T2

belongs to E. We have T > t and

on B

onQ\B [Xr] > Eft [XT.] for i = 1, 2. Thus

the collection

{EFt[X,.]:rEE,T>t} is directed. Apply (4.1.1(c)) to complete the proof of (2) in the case a = t.

4.1. Essential and stochastic convergence

125

In the general case, for each t E J with {a = t} # 0, choose a sequence T(t) E E with r,( V) > t such that

Zt = supEft [Xm] n

Now (4.1.23) there exist stopping times Tn with rn = have ZQ = Zt on {a = t}, so we get Zr,, = sup EF [XT ]

Tnt)

on {a = t}. We

.

n

(3) Let a1 < a2 be simple stopping times. Choose Tn > a2 with El 2 T Z. Now Tn > a1, so Z 1. So by the monotone convergence theorem, (Zt) is a submartingale, and E [Zol] > linmE [E-°1 [XTn]] = linmE [E"r'*°2

E[Z12]

As an application of the Snell envelope, we prove a stronger form of the

amart Riesz decomposition (1.4.6). Recall that an amart potential is an amart (Xn) such that limn E [Xn 1A] = 0 for all A E U°0=13 m. A Doob potential is a positive supermartingale Sn with limn E [Sn] = 0.

(4.1.15) Theorem. Let (Xn)nEIIV be an adapted process. Then (Xn) is an amart potential if and only if there is a Doob potential (Sn) with IXnI < Sn a.s. for all n. Proof. Suppose I Xnl < Sn a.s. Then IX < S, for all or E E. Now E [So] decreases as a increases. But E [Sn] -> 0, so also E [So] -> 0. Then IE [Xo] I < E [X01] < E [So], so (Xn) is an amart and E [I Xnl ] -> 0, so it is an amart potential. Conversely, suppose (Xn) is an amart potential. Then (I Xn I) is also an amart potential. Let (Sn) be the Snell envelope of the process (IXn1). Then (Sn) is an amart and a supermartingale (1.4.3). Also, Sn > IXnI, so Sn > 0. Finally, E [IXTI] -+ 0, so by (4.1.14(2)) we have E [Sn] 0. j

The Riesz decomposition and (4.1.15) indicate that for J = IN, the amart convergence theorem is not likely to have striking applications not possible with martingales and supermartingales. In the vector-valued case (Chapter

5) the situation is similar for uniform amarts (5.2.13), but not for other classes of amarts, since the Riesz decompositions are less restrictive. Also on directed sets (this Chapter) the behavior of amarts cannot be reduced to that of martingales and supermartingales.

Directed index set

126

Complements

(4.1.16) (Fatou's lemma for stochastic convergence.) Let (Xt) be a net of nonnegative random variables. Then E [slim inf Xt] < lim inf E [Xt] .

(4.1.17) (Uniform integrability and stochastic convergence.) Let (Xt) be a uniformly integrable net in L1. If Xt -> X stochastically, then E [Xt] E [X]. In particular, if Xt -* X essentially, then E [Xt] --> E [X]. (4.1.18) (Monotone convergence theorem for essential convergence.) Let (Xt) be a net bounded in L1 that is monotone increasing in the sense that if s < t, then X3 < Xt a.s. Then e lim Xt exists and lim E [Xt] = E [e lim Xt]. In particular, slim Xt exists and lim E [Xt] = E [slim Xt]. (4.1.19) If a net (Xt) converges in Lr norm, for some p (1 < p < oo), then (Xt) converges stochastically. (4.1.20) Let (Xt) be a net of random variables, and let A E R. Let

At={Xt>4}andBt={Xt>a}. Then:

{X3VXt>A}=A,UAt9B.UBt={X8VXt>A} {X3AXt>Al =A,nAt cB.nBt ={X8AXt>A} {Xt1 VXt2V...>A}=At,UAt2U...CC Bt, U Bt2 U ... C- {Xtl V Xt2 V ... > Al

{Xt1AXt2A...>A}cAt,nAt2n...c cBtlnBt2n...={Xt1AXt2A...>A}. (4.1.21) In the same notation, (a) less sup Xt > A} = ess sup At c_ ess sup Bt c_ less sup Xt > Al (b) less inf Xt > Al c_ ess inf At c ess inf Bt = less inf Xt _> A) (c) {e lim sup Xt > A} c e lim sup At c e lim sup Bt c {e lim sup Xt > Al (d) {e lim inf Xt > A} c e lim inf At c e lim inf Bt c {e lim inf Xt > A}.

(4.1.22) Let (Xt) be a net of random variables, and let A E R. Then slim sup{Xt > Al c_ {slim sup Xt > Al. t

t

Indeed, suppose Y is a random variable asymptotically above Xt, that is P{Xt > Y} -* 0. Write Bt = {Xt > A}. Then

P(Bt\{Y>A})=P{Y A on slim sup Bt. This is true for all such Y, so by the definition of slim sup Xt, we have also slim sup Xt > A on slim sup Bt. That is, {slim sup Xt > Al slim sup Bt.

4.2. The covering condition (V)

127

Note that in general the reverse inclusion slim sup{Xt > Al

{slim sup Xt > A}

-

t

t

fails. For example, with J = IN, if Xn = 1 - 1/n and A = 1, then the right side is 1 and the left side is 0. (4.1.23) (Generalized waiting lemma.) Generalize (1.1.5) to directed sets: Let (.Ft)tEJ be a stochastic basis, and let E be the corresponding set of simple stopping times. Let or E E be given, and for each t E J with for = t} # 0, let Tit) E E be given with T(t) > t on {a = t}. Then r defined by r(w) = r(t) (w) on for = t} belongs to E and r > or. Remarks

Stochastic upper and lower limits (also called upper and lower limits in measure) are due to D. E. Menchoff. The definition and treatment were considerably simplified by Goffman & Waterman [1960]. We have used their definition. Propositions (4.1.11) and (4.1.12) are from Millet & Sucheston [1980e]. For processes indexed by IN, the connection between maximal inequalities and convergence has been much studied; see for example Burkholder [1964]. Demiconvergence of martingales was first observed by Edgar & Sucheston [1981]. See also Millet & Sucheston [1983] and Frangos & Sucheston [1985].

4.2. The covering condition (V) Martingales indexed by a directed set converge stochastically. We will now provide an example showing that they need not converge essentially.

(4.2.1) Example. Let J be the set of all finite subsets of IN, ordered by inclusion. Then J is a countable directed set. Let Un be independent, identically distributed random variables with

P{Un = 1} = P{Un = -1} = 2. For each finite set t E J, let Ft be the (finite) a-algebra generated by the random variables Un, n E t. Thus, if s C t, then Fg C Ft. Define

Xt=1: 1Un. n nEt

First, we verify that the process (Xt) is L1-bounded. In fact, it is L2bounded: since the Un are orthonormal,

(1)2=

(1)2 < IIXtll2= nEt

nEIN

Directed index set

128

For Ll-boundedness, apply the Schwartz inequality:

liXtili = E [IXtI] = E [IXtI 1] < E [IXtl2]1/2E [12]1/2 = IIXtII2.

Next, to show that (Xt) is a martingale, we claim: if s C t, then Eye [Xt] = X3. Since t is s plus a finite number of extra elements, it is enough to consider the case where there is one extra element, t = s U {m}, and then apply induction. But Ur, is independent of .F.7 and therefore

Xe+mE[U,.]=X3.

E-F°[Xt]=Ey° [XB+mUml

is a martingale. This shows that Now we know by (1.3.1) that (Xt) converges stochastically. (In fact, by elementary Hilbert space theory, it converges in L2 norm and therefore by (4.1.19) stochastically.) But we claim that it does not converge a.s. Since J is countable, this means also that it does not converge essentially. In fact, we will see that the set of w E fI for which the net (Xt(w))tEJ converges has probability 0. Indeed, almost all w E S2 satisfy IU,,(w)I = 1 for all n.

Convergence of (Xt(w)) to x means: for any e > 0, there is a finite set s C IN such that for all finite sets t 2 s, <e. nEt

But this is equivalent to saying that of the series EnEIN(1/n)Un(w) converges absolutely. Now InUn(w)I=nF_

nEIN

H =°°,

so the series EnEIN (1/n)Un(w) does not converge absolutely. Therefore (Xt(w)) does not converge. Condition (V) Let (.F't)tEJ be a stochastic basis. Recall the notation E for the set of simple stopping times. For certain stochastic bases (for example a stochastic basis (.c'n)nEIN indexed by IN), all L1-bounded amarts converge essentially. For other stochastic bases (for example, the one in the preceding example) this is not the case. In this section we study a condition on the stochastic basis that will insure essential convergence. This condition is called the covering condition (V) or the Vitali condition (V). An adapted family of sets is a family (At)tEJ, where At E ft for all t E J. We will often write A* for e lim sup At. If r is a simple stopping time for the stochastic basis (.Ft), we consider a stopped set A(r) defined by

A(r) = U (At n {r = t}). tEJ

This union is finite, since {r = t} is empty except for finitely many values of t. It may be easily verified that A(r) E .F,.

The condition (V) states that the essential upper limit of an adapted family of sets may be approximated by a stopped set.

4.2. The covering condition (V)

129

(4.2.2) Definition. The stochastic basis (.Ft)tEJ satisfies the covering condition (V) if, for each adapted family (At)tEJ of sets, and each c > 0, there exists a simple stopping time r E E with P(A* \ A(T)) < E. To understand the meaning of this notion, recall that a.s. convergence of amarts was proved in Chapter 1 by approximation of lim sup Xt by the stopped process XT (1.2.4 and 1.2.5). On directed sets this is not possible in general. Condition (V) postulates such an approximation of e lim sup for zero-one valued processes (At). This assumption is crucial: it will be shown below that (V) is necessary and sufficient for convergence of L1-bounded amarts.

We have stated condition (V) in terms of stopping times since it will be used in that form. But it can also be stated without reference to stopping times: For each adapted family (At)tEJ and each e > 0, there exists an adapted pairwise disjoint family (Bt)tEJ, with only finitely many Bt nonempty, such that

P A* \ U(At n Bt) < e. t

Another variant allows countably many Bt nonempty, and concludes

P A*\U(AtnBt) =0. t

Another begins with a set B C A* and almost covers it by sets At n Bt. We will now prove the equivalence of a few simple variants of condition (V). To each variant there corresponds an asymptotic version in which, for each to, there is an approximating stopping time larger than to. The equivalence of an asymptotic version with the corresponding nonasymptotic version follows from the consideration of the process (Be) defined by Bt = (b)). Below, (b) is an At if t > to, Bt = 0 otherwise (see the proof of (a) asymptotic version of (a); conditions (c) and (d) are in asymptotic forms; their obvious nonasymptotic formulations have been omitted. Any of these equivalent formulations may be referred to as the covering condition (V).

(4.2.3) Proposition. Let (,Ft)tEJ be a stochastic basis. Then the following formulations of condition (V) are equivalent: (a) For every adapted family (At) and every e > 0, there exists T E E such that P(A* \ A(T)) < e. (b) For every adapted family (At), we have liminf, P(A* \ A(T)) = 0;

that is: for every e > 0, and every to E J, there exists r E E with T > to and P(A* \ A(T)) < e. (c) For every adapted family (At), we have limsup,,. P(A(T)) > P(A*);

that is: for every e > 0 and every to E J, there exists T E E with T > to and P(A(T)) > P(A*) - e. (d) For every adapted family (At), every to E J and every e > 0, there is a r E E with T > to such that P(A* L A(T)) < e.

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130

Proof. (a)

(b): Given to E J, define Bt as follows:

ift>to

(At

Bt

oth erwise.

0

Then B* = A*. By (a), there is T E E with P(B* \ B(T)) < e. Choose tl > T, tl > to. Define or by eT

=

ti

otherwise.

Then B(T) = B(v) C A(a), so P(A* \ A(a)) < E.

(b) = (c): By (b), there is T > to with P(A* \ A(T)) < e. Thus P(A(T)) > P(A*) - P(A* \ A(T)) > P(A*) - E. (c) = (d). Given e > 0, choose s E J, s > to, such that P ( ess sup At \ A*) < e. t>8

Let r > s be given by (c). Then P (A* L. A(T)) = P (A* \ A(T)) + P (A(T) \ A*)

< P (ess sup At \ A(T) I + P I ess sup At \ A* t>8

/

\ t>8

< P (ess sup At) - P (A(T)) + e t>8

< P(A*) - P(A(T)) + 2e < 3E. Clearly (d)

(a).

Condition (V) asserts that A* = e lim sup At may be "covered" by the stopped set A(T). We now show that (V) also holds if only a portion of A* can be covered, but a fixed portion, independent of the choice of (At).

(4.2.4) Proposition. Let (.Ft)tEj be a stochastic basis. Condition (V) holds if and only if there is a constant a, 0 < a < 1, such that for each adapted family (At) of sets there is a r E E such that P(A* fl A(T)) > aP(A*). Proof. Suppose a exists. Let (At) and E be given. Let r1 E E be such that P(A* fl A(T,)) > aP(A*). Then let s2 E J, 82 > Tl, and set

A' =

At \ A(Ti)

if t > sl otherwise.

Since A* \ A(Ti) = e lim sups Ai , there exists r2 E E such that r2 > s2 and

P((A*\A(Tl))flA(T2)) >aP(A*\A(Ti)). Then P(A*\(A(Tl)UA(T2)))
0; choose n so that (1 - a)nP(A*) < e. Choose s > Tn, and define

T=

on A(Tj) \ Uj=1 A(Tk), on 1 \ Uk=1 A(Tk).

Ti S

for 1 < j:5 n,

Then T E E and P(A* \ A(T)) < E. For the converse, suppose (V) holds; let a = 1/2 (say). Let (At) be given. If P(A*) = 0, then clearly P(A* flA(T)) > aP(A*) for any T. If P(A*) > 0,

use e = P(A*)/2 with condition (V) to obtain r with P(A* \ A(T)) < e. Then P(A* fl A(T)) < (1/2)P(A*). Example: Totally ordered basis Some simple examples may help explain the covering condition (V). The stochastic basis (.Ft)tEJ is totally ordered if, for any s, t E J, either .Fy C Ft

or Ft C .F3. Note that if J is totally ordered, then the stochastic basis is totally ordered, but the converse is not necessarily true.

(4.2.5) Proposition. If (.Ft)tEJ is totally ordered, then (.Ft) satisfies condition (V).

Proof. Let (At) be adapted and let E > 0. Then A* = e lim sup At C_ ess sup At, so there is a countable set {t}1 C J such that A* C U° 1 Ate Thus there is N E IN with P (A* \ UN 1 At,) < e. Renumber the ti so that C Ft,. Define T by: 'Ftl C J't2 C

1ti T=

ti tN

Then we have A(T)

on At, on Ati \ U,=1 Ate elsewhere.

for 2 < i < N

UN1 At,, and therefore P(A* \ A(T)) < e.

Example: Finite subsets of IN Consider the directed set J of finite subsets of IN, as in (4.2.1). We show now that condition (V) fails for the stochastic basis of the example (4.2.1).

See also (4.4.16), where it is shown that this basis also fails the weaker condition (C). We begin with independent, identically distributed random variables Un

with P{Un = 1} = P{Un = -1} = 1/2. If J is the set of all finite subsets

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132

of IN, ordered by inclusion, and .Ft is defined as the least v-algebra such that U,, (n E t) are measurable, then (.Ft)tEJ is a stochastic basis. If B, C are disjoint finite subsets of IN, write F(B, C) for the event

{UU,=1 forallnEB, U=-1 forallnEC}. Thus Ft has atoms F(B, C), where B n c = 0 and B U C = t. These atoms all have measure 2-k, where k is the number of elements of t. We claim that condition (V) fails. For m E IN, let Cm. be the set of all m-element subsets of {m + 1, m + 2, , 4m}. For m E IN and C E Cm, let t(m, C) = {1, 2, , m} U C E J. Define

At =

f F(C,0)

if t=t(m,C)for some mEINand CECm, otherwise.

0 1 We will show that (V) fails for the adapted family (At). First, we claim that e lira sup At = Q. Given s E J and e > 0, we can , m} and choose m so that s C {1, 2,

(3,m)

2-3m

(4.2.5a)

>

k =m

(This is a simple combinatorial lemma. A probabilistic proof is given in (4.2.19).) Then

U At 2 U F(C, 0), t>8

CECm

which is the event that at least m of the 3m random variables Um+1i , U4m are 1. Thus the probability is at least 1 - E by (4.2.5a). Um+2, Since s is arbitrary, this shows that e lim sup At = Q. Next, fix p E IN. Let T E E, r > {1, 2,- ,p}. Then

A(T)=U(Atn{T=t}) 00

=m=P UCECm U (F(C,0) n {T = t(m,C)}). Fix m > p. The atoms of Ft(m,C) contained in F(C, 0) have the form F(CUB, D) where BUD = {1, , m}, BnC = 0. Note P(F(CUB, D)) = 2-2m. Now two of the sets F(C U B, D), F(C' U B, D) are not disjoint, since they both contain F({m + 1, , 4m} U B, D). Thus, for a fixed pair (B, D), there is at most one C with F(C U B, D) n Jr = t(m, C)} # 0. So

P CUm (F(C, 0) n Jr = t(m, C) } f

<E Thus P(A(T))

2-2m = 2m 2-2m = 2-m

°°=P 2-m = 2-P+1. So condition (V) fails by (4.2.3(c)).

4.2. The covering condition (V)

133

Example: Interval partitions We now discuss a classical situation in which condition (V) holds (Theorem (4.2.8)). It is a result useful in derivation theory. The derivation of set-functions defined on Euclidean space Rd is closely related to an ap-

propriate directed set consisting of partitions. The topic is treated in a somewhat different setting in Chapter 7.

Let 1 be the d-dimensional cube [0,1]d, and let P be d-dimensional Lebesgue measure on Q. A collection C of open subsets of S2 will be called

substantial iff there is a constant M such that for every C E C there is an

open ball B with C C B and P(B) < MP(C). A simple example is the family d-dimensional intervals (rectangular solids with edges parallel to the axes) such that the ratio of the longest edge to the shortest edge is bounded by some constant M'. First, a Vitali style covering lemma: A stronger version of the Lemma is found in Chapter 7 (7.2.1) . If B is an open ball, we write r(B) for its radius.

(4.2.6) Lemma. (a) Let D be a collection of open balls in [0,1]d. Let W = U D. Then for each e > 0 there is a finite disjoint subcollection D' of D such that

E P(B) > 3-d (P(W) BED'

(b) Let C be a substantial collection of open sets in [0, 1]d with constant M.

Let W = UC. Then for each e > 0 there is a finite disjoint subcollection C' of C such that

P(C) > M-13-d (P(W) CEC'

Proof. (a) Let e > 0 be given. The set W is open, and therefore measurable.

So there is a compact set K C_ W with P(K) > P(W) - e. Now D is an open cover of the compact set K, so there is a finite subcover, say , Si,, E D and K C_ UP=1 Sj. Suppose these sets are ordered in decreasing order of their radii: r(Sl) > r(S2) > . . . > r(SP). , B,,, of balls. Let B1 = Now we define recursively a sequence 131, B2, , Bk have been defined. Let Bk+l be Sj, where j is the S1. Suppose 131, least index such that Sj fl Bi = 0 for 1 < i < k. If there is no such j, that is every Sj meets some Bi, then the construction stops with Bk. Certainly the construction stops in at most p steps. This completes the definition of

S1, S2,

the sequence 1 3 1 ,B 2 , .. . , B,,,.

Now for each Bi, let B' be the ball with the same center as Bi but three times the radius. We claim that P

n

j=1

i=1

US39UBZ.

Directed index set

134

Indeed, for each j, the ball S3 meets some Bi with r(Bi) > r(Sj), so S, C B. Thus

P(K) < P (U Sj) < P (U B:)
0

so small that M-13-d(1 - 2e) - e > 0. We will verify the condition in Proposition 4.2.4 with a = M-13-d(1 - 2e) - E. Let (At) be a family of sets adapted to the stochastic basis (.2) described,

and write A* = e lim sup At. We may assume P(A*) > 0. Write E' _ eP(A*). Choose s E J so that P(esssupt>3 At \ A*) < e'. There exists a sequence tk > s of indices with Uk At,, = ess supt>s At. Decompose each At,, into atoms Ck,,, of tk. Then P(Uk n Ckn) > P(A*). There is thus a

finite set F of pairs (k, n) such that P(UF Ckn) > P(A*) - e'. By the

lemma, there is a subset F' C_ F such that the atoms { Ckn : (k, n) E F' } are disjoint and P(UF1 Ckn) > M-13-d(P(UF Ckn) - e'). Now choose an index u larger than all tk where k occurs as a first coordinate in the finite set F'. By the disjointness of the Ckn with (k, n) E F', we may define a stopping time by if w E Ckn and (k, n) E F'

f tk u Thus A(T)

otherwise.

UF' Ckn. Finally,

P(A* n A(T)) > P(A(T)) - P(A(-r) \ A*) >M-13-d

(p( ICkn) -e1 'I -e' - e' J

> M-13 -d (P (A*) - 2,) = aP(A*).

4.2. The covering condition (V)

135

Essential convergence Here are some consequences of condition (V).

(4.2.8) Proposition. Let (.F't)tEJ be a stochastic basis. Suppose (V) holds. Then: (a) If (At) is an adapted family of sets, then slim sup A(T) = e lim sup At (= e lim sup A(T) rEE

\

tEJ

TEE

(b) If (Xt) is a stochastic process, then

slim supXt = elimsupXt I = elimsupX,) rEE

\

tEJ

rEE

That is, for every e > 0 and every to E J, there is a r E E with T > to and P11 e lim sup Xt - XT I > e l < E.

(c) If (Xt) is a nonnegative process, then for every A > 0 and every to E J, there is a r E E with T > to and P {e lim sup Xt > Al < 1 lim sup E [X,.] . A

TEE

(d) If (Xt) is a stochastic process, and v,,, is a sequence of simple stop-

ping times, then there exist T,,, E E with T,,, > on, and X,, -> e lim sup Xt a.s.

Proof. (a) Write A* = e lim sup At. To show slim sup, A(T) A*, we use (4.1.7a). If C is any set with lim, P (A(T) \ C) = 0, then we have

P (A* \ C) < P (A* \ A(T)) + P (A(T) \ C).

By (V), we have liminf, P(A*\A(T)) = 0. Therefore P(A*\C) = 0, or C A* a.s. This shows slim sup, A(T) J A*, so in fact slim sup, A(T) = A*. (b) We always have slim sup X, < e lim sup X, = e lim sup Xt, so we must prove the opposite inequality. Applying part (a), 4.1.21(c) and 4.1.22, we have for any \ > 0 { elimsupXt > A } C_ elimsup{Xt > A} tEJ l tEJ = slim sup {XT > A} JJJ

rEE

C { s lim sup Xr >,\ rEE

Therefore e lim sups Xt < s lim sup, XT.

I

Directed index set

136

(c) Let (Xt) be a nonnegative process, and let A > 0. Fix 3 with 0< Q < A. Define At = {Xt > 0}. Then {elimsupXt > /3} C A*. Let to E J and E > 0 be given. Then by (V), there is T E E with T > to and P(A* \ A(T)) < E. Then

E [Xr] _ E E [Xt 1{r=t}] t

/3EE [1{r=t}nA,] = /3P(A(T))

>,3(P(A*) -E) > ,3 (P{elimsupXt > i3} - E). Now to and E were arbitrary, so

P{elimsupXt >)31
to with

P{I elimsupXt - X,I > E} < E. Apply this recursively.

Convergence theorems hold in the presence of condition (V). Our proof will follow the method used in Chapter 1. A semiamart is a process (Xt)tEJ with lim sup IE [X,] I < oo. 'EE (See, for example, (1.4.26).) Clearly every amart is a semiamart. Note that for J = IN this definition is equivalent to supCEE IE [Xa] I < oo (Lemma (1.2.1)).

(4.2.9) Lattice property. (1) If (Xt) is an L1-bounded semiamart, then (Xt) is also a semiamart. (2) If (Xt) is an L1-bounded amart, then (Xt) is also an amart.

The proof is essentially the same as that of Theorem (1.2.2), and is therefore omitted.

(4.2.10) Theorem (Astbury). Let (Ft)tEJ be a stochastic basis. The following are equivalent: (1) Condition (V). (2) Lam-bounded amarts converge essentially. (3) L1-bounded amarts converge essentially.

4.2. The covering condition (V)

137

Proof. (1) (2). Let (Xt) be an L, -bounded amart. For each n E IN, choose to E J so that if r, or > tn, then IE [Xr] - E [Xo] I < 1/n. Then by Proposition (4.2.8(d)), there exist stopping times rn with rn > tn, Tn+l >rn, and Xr,. --i elimsupXt a.s. By Proposition (4.2.8(d)) applied to (-Xt),

there exist stopping times Qn with vn > tn, vn+1 > vn, and Xo -> e lim inf Xt. Hence 0 = lim J E [Xrn ] - E [XQn]

E [e lira sup Xt - e lim inf Xt],

so a lim sup Xt = e lim inf Xt.

(1) : (3). Suppose (1) holds. Then (as we just proved) also (2) holds. Let (Xt) be an Ll-bounded amart. By the lattice property (4.2.10), if A > 0, then the process ((-A) V Xt A A) is an Lam-bounded amart. Therefore by (2) it converges essentially. Therefore the original process (Xt) converges essentially on the set S1A = {elimsuplXtI < Al. But the maximal inequality (4.2.8(c)) shows that P{e urn sup IXtI < oo} = 1, so S2 is the countable union of sets Sia, hence (Xt) converges essentially. (3) (2)

(2) is easy.

(1). Let (At) be an adapted family of sets. For t E J, let Xt be

the Snell envelope (4.1.15):

Xt = esssupEFt [1A(r)] . r>t The net (XQ)CEE is decreasing, so (Xt) is an Lam-bounded amart. So by (2)

it converges essentially. Now Xt > 1At and lA* < e lim sup Xt, so by the essential convergence, e lim sup Xt = slim sup Xt, and thus by (4.1.11(c)),

P(A*) < P{elimsupXt > 1} < limsupE[XX]. rEE

Given or E E, there exists a sequence Tn > or such that E" [1A(-r,)] T Xo, so

E [Xo] = linm E {Ey' [1A(rn)] ] = linm P (A(7-.)) .

Hence limsupr E [Xr] < limsupr P(A(T)). Thus

P(A*) < limsupP(A(r)), r so (V) holds.

The next corollary is an immediate consequence of Theorem (4.2.11).

(4.2.11) Corollary. Let (Ft)tEJ satisfy (V). Then Ll-bounded martingales converge essentially

Note, however, that under condition (V), it it not necessarily true that Ll-bounded submartingales converge essentially (4.2.17). We will see in Section 4.4 that condition (V) is not necessary for convergence of Llbounded martingales.

Directed index set

138

Complements

(4.2.12) (v-directed set.) Suppose the directed set J has the property that every countable subset has an upper bound. (We say that J is a a-directed set.) Then any stochastic basis (.Ft)tEJ indexed by J satisfies condition (V). To see this, let (At) be adapted, and suppose a = lim inf P(A* \ A(T)) > 0. TEE

Next, choose indices sn E J with 1

inf P(A* \ A(T)) > a - n for n = 1, 2,. . . . There is soo E J larger than all sn, so

liminf P(A* \ A(T)) = inf P(A* \ A(T)) = a. TJ8

Now choose 7-1 = soo, and continue choosing recursively Tn with Tn '* Tn+1

so that P(A* \ A(Tn)) -* a. Choose too larger than all Tn. Then define a countably valued stopping time Too by: Tn

Too _ too

on A(Tn) \ elsewhere.

A(TE)

for n = 1, 2,

So A(To) _2 U A(Tn), and thus P(A* \ A(To)) = a > 0. But A* \ A(To) is the e lim sup of the At \ A(To), so there is t1 > to such that P ((A* n At,) \ A(Too)) > 0, so P (A* \ (A(TOo) U At 1))) < a. Then for large enough n we have also P (A* \

(A(Tn)UAt1))) < a. Then we may construct a E E with P(A* \ A(a)) < a, a contradiction. (4.2.13) (Condition (V).) All the conditions in Theorem (4.2.9) are equivalent with condition (V). (4.2.14) (Other generalizations.) Let (Xt)tEJ be a stochastic process. For u, ,r E E, a < T, write

H(a, T) = X, - E° [XT] . Then we say that (Xt) is a pramart if slim H(a, T) = 0; O 0 there exists a stopping time r E E controlled by an ordered stopping time v E E°, and a set B E Fp such that B C_ A(T) and

P(A\B)<e.

Then we have (VC) (V°) (V) and the implications are not reversible. (4.2.25) (Controlled amarts.) Write EC for the set of all controlled stopping times. If rl, T2 EEC, write rl to; we write T < to if It E r} = 0 except fort < to. If o, T E EM, we write o 0, there exists T E Elm with P(A* \ A(T)) < c and Ile(T)llw < e.

First we will record some useful technical variants of condition (Ve). Also, since P(A(r) \ A*) --+ 0 by the definition of a lim sup, any of the formulations with P(A* \ A(T)) < e may equivalently be restated with P(A* 0 A(T)) < e. Any of these formulations may be called "condition

Directed index set

146

(4.3.2) Lemma. The following are equivalent. (a) (V.,): for every adapted (At)tEJ and every e > 0, there exists T E Elm with P(A* \ A(T)) < E and IIe(T)II. < e.

(b) For every adapted (At)tEJ and every e > 0, there exists r E EM with P(A* \ A(T)) < e and IIe(T)IIw < e.

(c) For every adapted (At)tEJ, every e > 0, and every to E J, there exists ,r E EM with T >_ to, P(A* \ A(T)) < e and IIe(T)IIw < E.

(d) For every adapted (At)tEJ, every e > 0, and every to E J, there exists T E EIM subordinate to (At)tEJ with T > to, P(A(T)) > P(A*) - e, and IIe(T)II,, < e.

Proof. (a)

(b). Given T E EIM, choose s E J with s > r, and define T'

by

ItET}=

it ET} St\D(T)

t

s

t=s

.

Then T' E EM, A* \ A(T) At \ A(T'), and e(T) = e(T'). (b) (c). Given (At)tEj and to, define (Bt) by Bt = At if t > to and Bt = 0 otherwise. Then B* = At and we may apply (b) to the family (Be) to get (c) for the family (At). (c) (d). Given T E EM, let r' E Elm be defined by {t E T'} _ At n {t E T}. Then e(T') < e(T) and A* \ A(T) = A* \ A(T'). (d) (a). Let (At) be an adapted family, and lets > 0. Choose to so that P{ess supt>to At \ A* j < E/2. By (d) there exists r E Elm subordinate

to (At) with r > to, P(A(T)) > P(A*) - E/2, and IIe(T)IIw < e/2. But then P(A* \ A(T)) = P(A*)

- P(A(T)) + P(A(T) \ A*) < 2 + 2 = E.

It should be noted that if L1, = Lam, then condition (V.,) becomes condition (V). So (V) is sometimes known as (V,,.). Similarly, when L4, = Lp, we write (Vp) for (V4). We will see that under fairly general conditions, (Vw) is necessary and sufficient for the convergence of Lw-bounded martingales. The case of LIbounded martingales is excluded, but Lq-bounded martingales are included, for 1 < q < oo. There is a sort of convergence that is suggested by condition (V..). Suppose f (T) E IR for each T E EM. We will write V4,lim f (T) = u

if for every e > 0, there is to E J and 6 > 0 so that, if T E EM, T _> to, IIe(T) III < 6, then If (T) - uI < E. Note the use of EM, not Elm. This could be considered the Moore-Smith convergence defined by a directed set. The

4.3. Lw-bounded martingales

147

elements of the directed set are pairs (T, 6), where r E EM, b > 0, and II e(r) I I w

< b. The ordering is given by:

(r,b) < (T',s') if 6 > 6' and either r 4< r' or r = T'. Note that if Vwlim f (T) exists, then limrEE f (r) exists, and the limits are equal. There is a corresponding Vwlim sup. We will write

Vwlimsup f (T) < K

if for every e > 0, there exist to E J, and 6 > 0, such that if r E EM with r > to, IIe(r)IIt < 6, then f (T) < K + e. Clearly there is also a corresponding Vwlim inf. There are classes of processes that we may naturally associate with condition (Vw).

(4.3.3) Definition. Let (Xt)tEJ be an adapted process. Then (1) (Xt)tEJ is a Vw-amart if VwlimE [X] exists; (2) (Xt)tEJ is a V,-semiamart if Vplim sup IE [Xr] I < oo. (3) (Xt)tEJ is a V4,-potential if Vwlim E [I XT I] = 0.

(4.3.4) Proposition. If (Xt)tEJ is a Vw-amart and limt E [Xt 1A] = 0 for all A E Ut ft, then (Xt)tEJ is a V4,-potential.

Proof. A Ve-amart is an amart, so (Xt) is an amart potential. In particular, E [I Xt I ]

-> 0. Now given e > 0, choose b > 0 and to E J so

that if a1ia2 E EM, a1,a2 > to, IIe(a1)IIw < 6, and IIe(a2)IIw < 6, then IE [Xa1 - Xa2] I < e. Let or E EM with IIe(a)II4, < b and or > to. Choose t1 > a. Let A = Ut{Xt > 0,t E a}, so that A E Ft, Define rl,,r2 E EM by:

{tE}= t {Xt>O,tEa} Q\A ({Xt1ftEc') CI] IXtI 1{tEa} t

_ E (Xt 1{tEa,Xt>0} - Xt 1{tEa,Xt) if for each e > 0 there exists rl > 0 such that for each 7 > 0 and each adapted family (At)tEJ, there exists T E EIM with P(A* \A(T)) < e and E [4)(rle(rr))] < y. It can easily be verified that if 4) satisfies the (02) condition, then (D ,) is equivalent to (VO. Suppose J has a countable cofinal set. Then (D') holds if and only if every Lq,-bounded martingale converges (Talagrand [1986]). This result is more satisfactory than Theorem (4.3.11) in the sense that the hypothesis of condition (02) is not needed. But the covering condition (D.') is not as easy to understand or verify as condition (Vw). Talagrand introduced another covering condition (C..) in his study of Lw-bounded martingales (4.4.12). Remarks

The covering condition (Vp) appears first in the derivation setting; see, for example, Hayes [1976]. The martingale material here is based on Krickeberg [1956] and Millet [1978]. Krickeberg proved that (VP) implies essential convergence of all

L9 bounded martingales, where 1/p + 1/q = 1. He also proved the necessity in the case p = oo. (This is our (4.2.12).) See Krickeberg & Pauc [1963]. Following the lead of Hayes in the derivation setting, Millet proved in the directed-set setting necessity for (Vp), 1 < p < oo, as well as for (V..), where 4) satisfies condition (O2) (4.3.11). Millet & Sucheston [1979a] and [1980d] proved several equivalent formulations of (VP), including the amart convergence: they used the term "amart for Mr" for our "VP amart." These papers also introduced the stopping-time formulation of the Vitali conditions. Talagrand [1986] contains a theorem stating that condition is equivalent to the convergence of all LT-bounded martingales under a much more general hypothesis than (A2); unfortunately, we believe there is a gap in his proof. Professor Talagrand now shares this view. It would be interesting to know exactly which Orlicz functions have this useful property. The functional condition (FV4,) appears first in the important paper, Talagrand [1986]. Theorem (4.3.15) appears there.

4.4. Ll-bounded martingales

159

4.4. L1-bounded martingales In this section we return once again to the question of essential convergence of Ll-bounded martingales.

We know that condition (V) = (V,,.) is sufficient for convergence of Li-bounded martingales. We know that condition (V4.) is necessary and sufficient for convergence of L.,-bounded martingales, if 4) satisfies condition (02). But the Orlicz function 4 (see Section 2.2) does not satisfy (02). We will give an example below (4.4.10) showing that condition (V) is not necessary for convergence of Ll-bounded martingales. We will therefore need to consider another covering condition, condition (C), which is slightly more complicated than (V). Under reasonable hypotheses (the directed set J has a countable cofinal subset) we will prove that (C) is necessary and sufficient for the essential convergence of Libounded martingales. We will retain the notation used in the preceding section. A probability space (1, F, P), a directed set J, and a stochastic basis (Ft)tEJ will be fixed. We write Elm for the set of (simple) incomplete multivalued stopping times, and EM for the set of multivalued stopping times. If (Xt)tEJ is a stochastic process, then X* = e lim sup Xt t

and if -T E Elm,

Xr = > Xt 1{tEr} tEJ

If (At)tEJ is an adapted family of sets, then A* = e lim sup At t

and if -r E Elm,

A(T) = U ({t E r} n At). tEJ

For r E Elm, the domain of T is D(T) = Ut{t E T}, the sum of T is S(T) = E 1{tEr}, and the excess of r is e(T) = S(T) - 1Diri. We say T is subordinate to (At)tEJ if It c r} C At for all t E J; then A(T) = D(T). Covering condition (C) The next covering condition to be introduced is similar to the conditions

(V4.) of the previous section. But instead of requiring that the overlap should be small, it is only required that it be bounded.

(4.4.1) Definition. The stochastic basis (Ft)tEJ satisfies the covering condition (C) if, for each e > 0, there is a constant M such that for every adapted family (At)tEJ of sets, there exists r E EIM with P(A* \ A(T)) < e and S(r) < M. Note that (V) implies (C) even with M = 1. The stochastic basis of Example (4.2.1) fails condition (C): see (4.4.16). Here are some technical variants of condition (C). The proofs are similar to those used for (V4.) so they are left to the reader.

Directed index set

160

(4.4.2) Lemma. The following are equivalent. (a) (C): For every e > 0, there is M such that for any adapted family (At)tEJ of sets, there exists T E Elm with P(A* \ A(T)) < e and S(T) < M. (b) For every e > 0, there is M such that for any adapted family (At)tEJ of sets, there exists T E EM with P(A* \ A(T)) < e and S(T) < M.

(c) For every e > 0, there is M such that for any adapted family (At)tEJ of sets, and every to E J, there exists r E EM with r > to, P(A* \ A(T)) < e and S(T) < M. (d) For every e > 0, there is M such that for any adapted family (At)tEJ

of sets, and every to E J, there exists r E EIM, subordinate to (At)tEJ, with r > to, P(A(T)) > P(A*) - e and S(T) < M. (e) For every e > 0, there exist M and a > 0 such that for any adapted

family (At)tEJ of sets with P(A*) > e, and every to E J, there exists ,r E EIM, subordinate to (At)tEJ, with r > to, P(A(T)) > a and S(T) < M. See Millet & Sucheston [1980e], where the lemma is proved. The same paper contains a proof that the maximal inequality in Lemma (4.4.3), below, is equivalent to condition (C). Again, we refrain from studying the processes corresponding to condition

(C) and proceed directly to martingale convergence. However, our usual technique of truncating at A and -A to obtain an L..-bounded amart seems difficult to carry out here: we do not know of a suitable amart definition corresponding to condition (C). So we will have to use another method, involving a decomposition theorem for martingales (1.4.17). We begin with a maximal inequality. (4.4.3) Lemma. Suppose condition (C) holds. If (Xt)tEJ is a nonnegative adapted process, then for every e there exists M such that, for every A > 0,

sup E[X]. P(X*>A)<e+1A ,,EEIM S(o)<M

Proof. Let A > 0 be given. Let 0 satisfy 0 < )3 < A. Given e > 0, choose M as in condition (C). Let At = {Xt >,31. Then {X* > A} C A*. Then

there is r E Elm with P(A* \ A(T)) < e and S(T) < M. Then we have A(T) = Ut({t E T} n At), so 1

1A(r)

1At 1{tE-r}

1

E Xt 1{tE-r} = -X.r-

Thus P(A(T)) < (1/0)E [Xr]. Therefore P(X* > A) P(A*) < P(A(T)) +e

e+ )3E [X] <e+ 1 sup E[XQ]. a OEEIM S(o)<M

Nowlet,3-+A.

4.4. Ll-bounded martingales

161

(4.4.4) Proposition. Let (Ft)tEJ be a stochastic basis. Suppose condition (C) is satisfied. Then all uniformly integrable martingales converge essentially.

Proof. Let Xt = E-*)''t [X] be a uniformly integrable martingale. We may assume that X is o (U Ft)-measurable. Fix e > 0. Let M be the constant that corresponds to e by (C). There is s E J and bounded F8-measurable function Y such that IIX - YIII < e2/M. Now Zt = EFt [IX -YI] is a nonnegative martingale. Thus we have by the maximal inequality (4.4.3): there is T E Elm with r > s, S(r) < M, and

P{Z* > e} < 2E + E [Zr] Choose t > r. We have

P{Z* > e} < 2e + E [ZtS(T)] 2

< 2e + M E [Zt] < 2e + M M = 3e. Thus P(e lim sup Xt - e lim inf Xt > 2e)

< P ({e lim sup Xt - Y > e} U {e lim inf Xt - Y < -e}) < P{Z* > e} < 3e. Thus Xt converges essentially. Since Xt converges stochastically to X, the essential limit is also X. Next, we must handle the case of nonuniformly integrable martingales. Recall these definitions: A finitely additive set-function p defined on a sub-

algebra G of F is called singular if for every e > 0, there exists A E 9 with P(St \A) < e but variation IAI(A) < e. A martingale (Xt)tEJ is called singular if the finitely additive set-function p defined on = Ut.F't by p(A) = lim E [Xt 1A]

is a singular measure. We know (1.4.17) that any Ll-bounded martingale can be written as the sum of a uniformly integrable martingale and a singular martingale.

(4.4.5) Proposition. Let (.Ft)tEj be a stochastic basis. Suppose condition (C) is satisfied. Then all singular Ll-bounded martingales converge essentially to 0.

Proof. Suppose (.F't)tEJ is a singular Ll-bounded martingale. If (Xt) corresponds to singular measure u: µ(A) = lim E [Xt 1A]

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162

for all A E G = Ut.Ft, then the absolute value process Zt = XtI corresponds to the variation of p:

1pl(A)=limE[Zt1A]. Thus p has bounded variation, since (Xt) is Ll-bounded. (The process (Zt) is a submartingale.) Let e > 0. Let M be the constant corresponding to e by condition (C).

Since p is singular, there exist s E J and B E F. with P(B) > 1 - e and 1i1(B) < e2/M. If a E Elm with a > s and S(a) < M, then choose t > a and compute E [Zo 1B] < ME [Zt 1B] < e2.

Now by the maximal inequality (4.4.3), applied to the process (Zt 1B)t>8,

we have P{Z* 1B > e} < 2e, so P(Z* > e) < 2e. Thus Z* = 0, so that (Xt) converges essentially to 0.

(4.4.6) Theorem. Let (Ft)tEJ be a stochastic basis. Suppose condition (C) is satisfied. Then all Ll-bounded martingales converge essentially.

Proof. Apply the decomposition (1.4.17), then use Theorems (4.4.4) and (4.4.5).

Necessity of (C)

We next undertake the proof of the converse: If all Ll-bounded martingales converge essentially, then condition (C) holds. (We assume the existence of a countable cofinal set for this.) In fact, we need only the convergence of the uniformly integrable Ll-bounded martingales; that is, martingales of the form Xt = E.Ft [X] for some X E LI. We will say an adapted family (At)tEJ of sets is finite if At = 0 except for finitely many t. (Such an adapted family is thus really the same thing as an element of Elm, viewed from a different perspective.) We also say that the family (At)tEJ is supported beyond to if At = 0 for all t except those with t > to. (Thus the corresponding r E EIM satisfies r > to.) We begin with an application of the Hahn-Banach theorem.

(4.4.7) Lemma. Let (At)tEJ be a finite adapted family of sets. Let a > 0. Suppose, for each adapted family (et)tEJ of nonnegative bounded functions, subordinate to (At)tEJ, with E [ E et] = 1, we have a. 00

Then, for each y > 0, there is Y ELI with Y > 0, E [Y] < 1/a, and

P (At \ {EFt [Y] > 1/2}) < y for all t.

4.4. Ll-bounded martingales

163

Proof. Fix ry > 0. Consider the following two subsets of Lam:

Cl = {

:

= E, for some family (et)tEJ of nonnegative functions, subordinate to (At), E

1,

,0

< 1/7 }

C2={ ELF:. 2

for all

E C2

for all

E C1.

Now E 1/2 for all < 0 implies Y > 0; and E 1/2 for all with a/2 implies IIYIII 1/2}. If P(B) > ry, then

= P(B) 1B

i
0, there exist to E J and N > 0 so that for all finite adapted families (At)tEJ of sets supported beyond to, with P(Ut At) > e, there is an adapted family (&)tEJ of bounded nonnegative functions, subordinate to (At)tEJ, with E [ E fit] = 1 and > & < N. Proof. Suppose (for purposes of contradiction) that this is false. Then there

is e > 0 such that for each to E J and N > 0, there is a finite adapted family (At)tEJ supported beyond to, with P(Ut At) > e such that any (t)tEJ subordinate to (At)tEJ with E [ E et] = 1 has 11 E t11. > N.

Directed index set

164

Let (sk) be an increasing sequence cofinal in J. For each k, use N = 2k+2/e to obtain a finite family (At)tEJ supported beyond sk as above. Say At # 0 for p different values of t. (Note p depends on k.) Then apply Lemma (4.4.7) with 'y = 1/(kp) and a = N to obtain Yk E Ll with IlYkjil < e/2k+2 and P (At

\ {EFt [Yk] > 2 })

gyp, 1

so that P

Let X =

U{E?t[Yk]>2} >.

Yk. Then X11l < e/4. Consider the uniformly integrable

martingale Xt = E-7t [X]. Since (sk) is cofinal, we have

P(e lim sup Xt > 1/2) > e.

But P(eliminfXt > 1/2) < e/2, since 1IXIIl < e/4. So (Xt) does not converge. This contradiction completes the proof.

(4.4.9) Theorem. Suppose J is a directed set with countable cofinal subset, and (.Ft)tEJ is a stochastic basis. If every uniformly integrable Li-bounded martingale converges essentially then (Ft)tEJ satisfies (C).

Proof. We claim: For every e > 0, there exist M > 0, a > 0, ry > 0, and to E J such that for any finite adapted family (At) supported beyond

to with P(U At) > e and E P(At) < (1 +'y)P(U At), there is T E Elm, subordinate to (At) with P(A(T)) > a and S(T) < M. Indeed, given e > 0, let N and to be as in Lemma (4.4.8), and let a = 1/(4N), ry = 1/(2N), and M = 4N. Let (At)tEJ be a finite adapted family supported beyond to with P(U At) > e and E P(At) < (1 + ry)P (U At). Overlap occurs on the set

C = U(As fl At). $34t

(It is really a finite union.) Then P(C) _< >P(At) - P(UAt) _< 7 = 1/(2N). There is a family (t)tEJ subordinate to (At)tEJ with E [

fit] = 1

and > t < N. Now E [(E t) 1c] < NP(C) < 1/2, so E

Thus if H = { by It E r} =

lO\c] >

2

t > 1/4} \ C, we have P(H) > 1/(4N). Define r E Elm 1/4}. Then, since the sets At do not overlap outside C,

4.4. Ll-bounded martingales

165

we have P(A(T)) > P(H) > 11(4N) = a. Also, S(T) < 4 E et 0 and let (Bt) be an adapted family of sets with P(B*) > E.

By (Vi), there is a finite adapted family (At), subordinate to (Bt), with P(U At) > e and > P(At) < P(U At) + ye < (1 + y)P(U At). So there is T E Elm with P(A(T)) > a and S(T) < M. By Lemma (4.4.2), we see that condition (C) is satisfied. A counterexample

(4.4.10) We will next consider an example of a stochastic basis (Ft)tEJ not satisfying condition (V), but for which all Ll-bounded martingales converge essentially.

If we wish, the probability space (Q,F, P) may be [0, 1) with Lebesgue

measure, and the sets Dt and It (W) defined below may be taken to be half-open intervals [a, b). But any continuous probability space will suffice.

Choose integers ni with 2 < ni < n2 < . . and E 1/ni < oo. Choose positive numbers ai with 1/2 > al > a2 > . . . and > ai < oo. For each i, let K(i) = {1, 2, , nil, and let L(i) be the collection of all 2-element subsets of {1, 2,. .. , nil. Thus L(i) has ni(ni - 1)/2 elements. Let Ji be the Cartesian product KI x K2 x ... x Ki, and let J = U000 Ji. [Write 0 for the unique element of J0.] For t E Ji, we say that t belongs to level i, and write Itl = i. If t = (ti, t2i , ti) E Ji and p E K(i + 1), then we write tp for the element of Ji+i defined by tp = (tl, t2i , ti, p). The set J is directed when the ordering is defined by:

s 0, that 9i, .Ft, and Dt (for Iti = i) have been defined.

Let t E Ji. Subdivide Dt into ni+l sets Dtp (p E K(i + 1)), each of probability (1 - al)(1 - a2) ... (1 - ai)(1 - ai+l)

nin2...nini+l

4.4. Ll-bounded martingales

167

and ni+1(ni+l - 1)/2 sets It(W) (W E L(i + 1)), each of probability (1 - al)(1 -

ai)ai+l 2

nln2.nini+1(ni+i - 1)

This subdivision is possible, since the sum of the probabilities of all the subdividing sets listed is exactly the probability (1 - a1)(1 - a2)...(1 - ai)

nin2...ni

of Dt. For p E K(i + 1), let

Atp=DtpUU{It(W):W E L(i+1),pEW}. Then

P(Atp) =

(1 - a,)...(1 - ai)ai+12

(1 - al)... (1 - ai+i)

+ (ni+1 - 1) nl ... nini+l(ni+l - 1)

nl ... ni+1 1 + ai+1

P(Dt)
i+1

so A* = limsupAt = f°_°o Ult1=i Dt. Now

P(A*) =i- lim P U Dt 00 ItI=i

= lim 00

/

(1-a,)...(1-ai)nl...ni nl ... ni

=fl(1-aj)>0. j=1

Directed index set

168

Now fix io and let T E E be any simple stopping time with values in levels io and above. Now At is an atom of Ft, so either {T = t} 2 At or {T = t} n At = 0. Because of the overlap properties of the At, for each fixed index t we must have Jr = tp} n Atp = 0 except for at most one value of p, so that

P

U {T = t} n At I< E P(Atl) Itl=i+1

/

Itl=i

< nl ... ni

Thus

00

2

ni+1

P(Dt)

1)

P(A(T)) < E n. i=zo

Since this approaches 0 as io --> oo, we see that (V) fails. However, we claim that all L1-bounded martingales adapted to (.Ft)tEJ converge essentially (or, what is the same thing since J is countable, a.s.).

We verify this by proving that condition (C) holds. In fact, a uniform version of (C) is true: If (Bt) is an adapted family of sets and e > 0, then there is T E ELM with P(B* \ B(T)) < e and S(T) < 2. Let (Bt) and s be given. Fix i. We will construct Ti E ELM with values in Ji such that S(Ti) < 2 and P (Ultl=i Bt \ B(Ti)) < 2/ni. First consider the atoms of gi_1. For s E Ji_1, if D3 C_ Bt for some t E Ji, then choose one and call it t = ry(s). For Isi < i - 2 and W E L(i - 1), if I3(W) C_ Bt for some t E Ji, then choose one and call it t = /3(s, W). If some atom D3, S E Ji_1, is not contained in any Bt, then consider the sets Hp = D8 nB,p. This is JFp measurable, and does not include all of Ds. So it is one of: 0, A8P, Dsp \ Asp. For this value of s, if Hsp = Dsp \ Asp for some value of p,

then choose one, and write Rsp = Hp for that value of p and Rsp = 0 for the other values of p. If Hp is 0 or Asp for all p, then let Rsp = Hsp for all

p. Now defineri by: It ETi}=0if Itl #i, and

It ETi}=U{Ds:t=ry(s)}UU{I3(W):t=,3(s,W)}URt. Clearly It E Ti} C Bt. Since the sets Rt intersect at most two at a time (and the others are disjoint), S(Ti) < 2. Now how large is Vi = Ultl=i Bt \ B(Ti)? Its intersection with most

atoms of gi_1 is 0. The only exception is an atom Ds, s E Ji_1, where Hsp = Dsp \ Asp, so P(V n Ds) 0, choose io with F°_io 2/ni < e/2, then choose it > io so that

P B*\ U Bt

E be a random variable. The Pettis

IIXIIp =up E [I (X, x*) I ] z' EE'

1Ix*II Li (1, F, P) by

T(x*) = (X,x*). We claim that the graph of T is closed. If x* -> x* in E*, then (X (w), x*) -> (X (w), x*) for each w, hence (X, x*) --> (X, x* ) in probability. If, in addition, (X, x*) -> Y in L1 norm, then (X, x*) -> Y in probability. Therefore Y = (X, x*) a.s. Thus T is a closed linear transformation defined on a Banach space, so it is bounded. The norm IITII is the Pettis norm of X. When the Banach space is separable, there are many equivalent ways to recognize random variables. Our proofs are based on a simple lemma. A closed half-space in a Banach space E is a set of the form

Ha(x*)={yEE: (y,x*) 1- 2-". j=1

Let aj E Dj and write Aj = {X E D. I. Then

Xn=Eaj1Ai M j=1

is a measurable simple function satisfying

P{JJX - Xnll < 2-n} > 1 - 2-n. Clearly X is Bochner measurable.

(5.1.7) Pettis measurability theorem. Let E be a Banach space, and let (St, F, P) be a complete probability space. A function X : Il -+ E is Bochner measurable if and only if X is scalarly measurable and there is a separable subspace El C E with P{X E E1} = 1. Proof. If X is Bochner measurable, then it is scalarly measurable and (almost) separably valued, since simple functions have these properties and they are preserved by pointwise a.s. limits. For the converse, suppose X is scalarly measurable and has values (a.s.)

in the separable space E1. By ignoring a set of measure zero, we may assume that X has all its values in El. Now the collection of all sets D C E1 such that {X E D} E F is a a-algebra and includes the closed half-spaces. Therefore it includes all Borel sets. This means that X is Borel measurable, so by the preceding result X is Bochner measurable.

A slight variant of this theorem will also be useful. If F is a Banach space and X : 1 -+ F*, we will say that X is weak-star scalarly measurable if (y, X) is a measurable function on f for each y E F.

Banach-valued random variables

176

(5.1.8) Proposition. Let F be a Banach space with separable dual, and let X : St - F* be weak-star scalarly measurable. Then X is Bochner measurable.

This is proved by almost the same method as we have used for the Pettis measurability theorem. We must use weak-star closed half-spaces in place of closed half-spaces. The unit ball of F* is weak-star closed, so it is the intersection of a family of weak-star closed half-spaces. The rest of the proof is the same. Vector measures

Let E be a Banach space. Suppose F is a a-algebra of subsets of a set Q. A vector measure in E is a function p: .F --f E such that

(QA)n = n-i

n-i

/t(An),

for every pairwise disjoint sequence (An) in .F, where the series converges in the norm of E. The measure p is absolutely continuous with respect to

P iff p(A) = 0 for all A E .F with P(A) = 0. (We write µ « P.) The variation of p on a set A E .F is n

I,u I (A) = sup i=1

IIp(Ai)II,

where the supremum is taken over all finite disjoint sequences Al, A2, , An C A in T. The set-function is a (possibly infinite) measure on.F. We say that p has a-finite variation on a set A if A is a countable union of sets on which p has finite variation. Suppose the random variable X is Bochner integrable, and a set function p is defined by p(A) = E [X 1A] for A E.F. Then p has finite variation; in fact the variation of p on St is exactly the Bochner norm E [IIXIII The semivariation of a vector measure µ on a set A E .F is IIpII(A) = sup { I x*,ul (A) : x* E E*, IIx* II < 11 ,

where Ix*,uI is the variation of the real-valued measure x*p (see Section 1.3). The set-function is not additive, but it is subadditive and has the following property: IIpli(A) = 0 if and only if p(B) = 0 for all B C A. Indeed, we claim that sup{ IIp(B)II : B C A} E such that µ(A) = E [X 1A]

for all A E .F. The random variable X is called the Radon-Nikodym derivative of µ with respect to P, and is denoted

XdP Here is an elementary, but useful, reformulation of the condition.

(5.1.11) Proposition. Let E be a Banach space. Then E has the RadonNikodym property if and only if for every probability space (0,.F, P) and every measure p: F -+ E such that II p(A) II P(A) for all A E F, there is a Radon-Nikodym derivative dµ/dP and Ildµ/dPII < 1 a.s. Proof. If IIp(A)II < P(A) for all A C -.F, then p has variation at most 1. If E has the Radon-Nikodym property, then it satisfies the condition stated here. Conversely, suppose E satisfies the condition. Let (52, F, P) be a probability space, and u: Y -+ E a measure, absolutely continuous with respect to P, with finite variation. We may suppose that µ is not identically 0. Define __

Iµ1(A)

P'(A) IpI( l) for all A E F. Then (Sl,F, P') is a probability space. Define

1µN) for A E.F. Then /,t': F --+ E is a vector measure, and II p'(A) I < P' (A) for all A E.F. Thus there is a Bochner integrable random variable X': Il -+ E such that

A '(A) =E'(X'IA) for all A E F, where E' is the expectation with respect to P'. Now P' is absolutely continuous with respect to P, so there exists a scalar-valued Radon-Nikodym derivative H = dP'/dP. A short calculation shows that the product X = lµI(S2) H X' is the required Radon-Nikodym derivative of µ with respect to P. There are some special cases under which Radon-Nikodym derivatives exist regardless of the Radon-Nikodym property. One of them occurs when the probability space (Il, P) is atomic. The Radon-Nikodym derivative of µ is

X = Z'P( A) 1A A

where the sum is over a maximal disjoint partition of f into atoms. Another such special case is the conditional expectation, which we consider next.

5.1. Vector measures and integrals

(5.1.12)

179

Let E be a Banach space, let (1, F, P) be a

Definition.

probability space, let X : 1 --+ E be a Bochner integrable random variable, and let g g F be a a-algebra. The conditional expectation of X given g is the unique (a.s.) random variable Y: St -+ E, measurable with respect to such that E [Y 1A] = E [X 1A]

for allAE9. We write Y=EO[X] or E [X 191. (5.1.13) Proposition. Let E be a Banach space, let (Q, .F, P) be a probability space, let X : St - E be a Bochner integrable random variable, and let 9 C_ F be a a-algebra. Then the conditional expectation EO [X] exists.

Proof. The conditional expectation V [X] exists for simple random variables of the form n

X=

1Aj xj, j=1

where A j E F and x j E E, namely n

E9 [X]

= EEg [1A,] xj j=1

It is not hard to establish the estimate E [IJE9 [X1] - E9 [X2] III < E [IIX1 - X2II] ,

where X1 and X2 are simple functions. If X is Bochner integrable, and (Xn)nEIN is a sequence of simple functions that converges to X in the Bochner norm, then the sequence E9 [Xn] is Cauchy in the Bochner norm. The limit of this sequence defines E9 [X]. The conditional expectation is a kind of "average," so the following result is not unexpected.

(5.1.14) Proposition. Let X be a Bochner integrable random variable taking values in a closed convex subset C of the Banach space E. Then (a) for any sub-or-algebra 9 of F, the conditional expectation E9 [X]

has values in C; and (b) for any A E F with P(A) > 0, we have E [X 1A] /P(A) E C.

Proof. Since X is Bochner measurable, its values lie a.s. in a separable subspace of E. So we may assume E itself is separable. By Lemma (5.1.5), C is an intersection of countably many closed half-spaces. That is, there exist linear functionals xi E E* and scalars ai such that n00

(5.1.14a)

C=

I

I

i=1

{ x E E : (x, xi) < ai } .

180

Banach-valued random variables

Now X has its values in C, so for any A E F we have (5.1.14b)

(E [X 1A] , xi) = E [(X, X7) 1A] < E [ai 1A] = ai P(A).

Therefore, by (5.1.14a), the vector E [X 1A] /IP(A) belongs to C. This proves (b). For the proof of (a), write Y = EO [X]. In (5.1.14b) take A= (w E Q: (Y(w), xi) > ai }. Then (E [Y 1A] , xi) = E [(Y, X7) 1A] > E [ai 1A] = ai P(A).

with equality only if P(A) = 0. But A E 9, so E [Y 1A] = E [X 1A], and equality holds. Thus P{(Y, xi) > ail = 0. The union of countably many events of probability zero still has probability zero, so by (5.1.14a), almost all the values of Y lie in C. A corollary of this is the vector-valued version of Jensen's inequality. (For a simplified statement, replace "lower semicontinuous" with "continuous." Recall that the function cp is called lower semicontinuous if, for each c, we have cp(c) < lim inf.,_,, cp(x).)

P) a probability (5.1.15) Theorem. Let E be a Banach space, space, 9 C_ F a v-algebra, and X : S2 -> E a Bochner integrable random variable. Let C C E be a closed convex set, and cp: C -* IR a convex lower oo, then semicontinuous function. If X E C a.s. and E cp(E4

[X]) < Ell [cp(X)]

a.s.

Proof. In the Banach space k = E ®1R, let C = { (x, t) : x c C, t > cp(x) }. Then C is a convex set since cp is a convex function and C is closed since cp is lower semicontinuous. Define X : Q -> E by k (Lo) = (X (w), cp(X (w) )) .

Then k isBochner integrable, with values in C, so the conditional expectation E[X 19] also has values in C. But

E9 [k] = (E9 [X] , E9 [(X)]), so we have EO [cp(X)] > cp(Eg [X]) a.s.

Jensen's inequality shows clearly that the conditional expectation is a contraction on the vector-valued LP spaces. If p > 1 then the function cp(x) = IIxiiP is convex, so

IlE [X]

IIP

< Ell [IIXIIP]

a.s.

Integrate both sides and raise to the power 1/p to obtain E [IIE11 [X] IIP]

1/p < E [IIXIIP] 1/P

(For the corresponding result in Orlicz spaces, see (5.1.22)).

181

5.1. Vector measures and integrals

We include here a few examples that illustrate the Radon-Nikodym property. The Banach space 11 is the set of all sequences (x1) x2, x3, ... )

of real numbers with

00 i=1

Ixil < oo.

1

It is a Banach space when given the norm E I xi I. The Banach space co is the set of all sequences (xl,x2,x3,...)

of real numbers with Jim 2-,00

xi = 0.

It is a Banach space when given the norm maxi I xi I.

(5.1.16) Proposition. (a) The space 11 has the Radon-Nikodym property. (b) The space co fails the Radon-Nikodym property.

P) be a probability space, and let µ:.F -* 11 satisfy 11p(A)II < P(A) for all A E.F. When p(A) is written in terms of its Proof. (a) Let components, µ(A) = (µ1(A), µ2(A), µ3(A), ... ),

each µi(.) is a measure (since the map that selects the ith component is linear and continuous). Each is a scalar-valued measure, absolutely continuous with respect to P, so there exists a Radon-Nikodym derivative Xi : SZ -> Ift such that pi (A) = E [Xi 1A]

for all A E F. Now E [IIXili] = Iµil(Q), and E°_1 lpil(S2) < oo, so for almost all w E S2, the combined random variable

X (w) = (X1 M, X2 M, X3 M, ... ) has its values in 11. This combined random variable satisfies

µ(A)=E[X1A] for all A E F. (b) Let (1k, F, P) be [0,1] with Lebesgue measure. Define a: F -* co by

p(A) = I

\ JA

sin w dw,

JA

sin 2w dw,

JA

sin 3w dw,

I .

///

By the Riemann-Lebesgue lemma, µ(A) E co for all A E .F. Also,

jjµ(A)jj < P(A)

182

Banach-valued random variables

for all A E.F. But we claim that u has no Radon-Nikodym derivative with respect to P. As in part (a), the continuity of the coordinate functionals on co shows that if a Radon-Nikodym derivative X did exist, we would have X (w) = (sin w, sin 2w, sin 3w,

)

for almost all w E [0, 1]. But this X(w) is in co for (almost) no w. This shows that co fails the Radon-Nikodym property. Complements

(5.1.17) (Banach space Li.) The space L1 = L1([0,1]) fails the RadonNikodym property. This can be seen using the probability space (Q, .F, P) =

[0,1] with Lebesgue measure and the vector measure p: F -> L1 defined by:

µ(A)=1A

for allAE.F.

(5.1.18) (Pettis integral.) There is another kind of integral sometimes used for random variables with values in a Banach space. Suppose the random variable X is scalarly integrable. We say that X is Pettis integrable on the set A E F if there is a vector XA E E satisfying

x*(XA) = E [x*(X)1A] for all x* E E*. We say that X is Pettis integrable if X is Pettis integrable on each set in F. The vector XA is called the Pettis integral of X on A, and we will usually write it using the same notation as the Bochner integral:

xA=E[X1A]. (5.1.19) (Pettis vs. Bochner.) Let X be a Bochner integrable random variable. Then X is also Pettis integrable, and the two integrals agree. (5.1.20) (Pettis vs. Bochner.) Define a random variable X: [0, 1] -> 12 as follows. Let en be an orthonormal sequence in 12, and let An be disjoint sets in [0, 1] with Lebesgue measures P(An) = 2-n f o r n = 1, 2, 3, . Define n

n 1An (w) en.

X (W) _ n=1

Then X is Pettis integrable but not Bochner integrable. (5.1.21) (Variant definition.) The Banach space E has the RadonNikodym property if and only if for every probability space (Sl, F, P) and every vector measure p: .F -* E with a-finite variation, absolutely continuous with respect to P, there is a Pettis integrable random variable X such

that

µ(A)=E[X1A] for all A E F.

5.2. Martingales and amarts

183

(5.1.22) (Vector-valued Orlicz spaces.) Let (F be an Orlicz function. For E-valued random variables, let

=inf{a>0:E['t (Ilallll E is a vector measure, then its (F-variation of µ with respect to P is sup

'D

(JJa(Ai)IJ

P(Ai)

)

P(Ai),

, An) in where the supremum is over all finite disjoint sequences (Al, A2i F. If X is a Bochner integrable random variable, and µ(A) = E [X 1A] for all A, then the (F-variation of p is equal to the Orlicz modular E [(F (I I X I I )1 of X.

Remarks

Additional material covering measurable subsets of a Banach space is in Edgar [1979b] and Talagrand [1984]. A more thorough discussion of vector-valued measures and the Bochner integral

can be found in Diestel & Uhl [1977], or Bourgin [1983]. These two books also contain much more material on the geometry of Banach spaces with the RadonNikodym property.

The proof [Proposition (5.1.16(a))] that 11 has the Radon-Nikodym property actually shows that any Banach space with a boundedly complete basis has the Radon-Nikodym property. (For the definition, see for example Lindenstrauss & Tzafriri [1968], page 13.) This fact is a special case of the result proved below (5.3.32) that a separable dual Banach space has the Radon-Nikodym property.

5.2. Martingales and amarts In this section we begin the discussion of martingales and related processes with values in a Banach space. Then we discuss some difference properties and Riesz decompositions for vector-valued processes. Convergence theorems will be treated in the next section, because of the close connection with the Radon-Nikodym property. Let (Il, F, P) be a probability space, let (Fn)nEiNV be a stochastic basis on S2, and let E be the set of all simple stopping times for (FF)nEJN. Let E be a Banach space, and let (Xf)fEN be a sequence of Bochner integrable random variables with values in E, adapted to (Fn)nE>rr. These data will be fixed throughout Section 5.2.

Banach-valued random variables

184

Elementary properties

(5.2.1) Definition. The E-valued adapted process (Xn)nEIN is a martingale if the net (E [Xo])oEE is constant.

One shows (applying a linear functional, then using the scalar case (1.4.3)) that (Xn)nErr is a martingale if and only if

EF [Xn] = X.

for m < n. Also, if (Xn)nE]N is a martingale, and or < r in E, then V'° [Xr] = X. The optional sampling theorem for martingales follows from this.

(5.2.2) Optional sampling theorem. Suppose (Xn)nEIN is a martingale

with respect to the stochastic basis (.Fn)nEIN Let rl < r2 < ... be an increasing sequence in E. Then the process Yk = Xrk

is a martingale with respect to the stochastic basis (Gk)kEN defined by 9k = .Trk .

There are several different vector-valued analogs of the scalar-valued amarts. One that inherits most of their properties is the "uniform amart."

(5.2.3) Definition.

The process (Xn)nEIN is a uniform amart if the

following difference property is satisfied: For every e > 0, there is mo E IN such that for all or, r E E with mo < or < -r,

E

[IIE-c- [Xr]

- XoII ] 0, there is mo E IN such that

IIE"*°[XT]-XuIIP<E

for all o,TEE withmo k. But

IIE [YXn]-E[Y'Xn] 11 < 1a=E. a Then we have for n, m > k IIE [YXn] - E [YX,n,] II < 3E,

so limn E [YXn] exists.

Now consider A E F. Then E [1AXn] = E [E-°° [1A] Xn]

so it converges as n -> oo. Let µ(A) be the limit. We must show that a is countably additive. Suppose Ak

0.

Given

e > 0, there is 6 > 0 so that P(D) < 6 implies IIE [1DXn] II < e. Now P(Ak) -> 0, so P(Ak) < 6 for k large enough, and thus IIE [lAkXf] II < E for all n. Therefore Ilµ(Ak)II < e. This shows limk IIp(Ak)II = 0. Thus µ is countably additive. The preceding result is also true for weak sequential amarts. Next we consider condition (B), namely sup(EEE [IIXoII] < oo. It was not necessary to emphasize condition (B) in the scalar case, because an Ll-bounded scalar amart automatically satisfies condition (B). This is no longer true for vector-valued amarts (5.5.29). However it does remain true for uniform amarts.

(5.2.18) Proposition. A uniform amart is Ll-bounded if and only if it satisfies condition (B).

Proof. By the Riesz decomposition (5.2.13), it is enough to prove the result when (Xn) is a martingale. But then (IIXnhi)nEIN is a submartingale. For every v in E there is an n E IN with v < n, and thus E [IIXoII ] < E [IIXnII ] . Therefore supE [IIXoII] = sup E [IIXnII] (EE

nEIN

The following is obtained by applying (1.1.7) to the real process (IIXnII) .

(5.2.19) Maximal inequality. Let (Xn)nEIN be an adapted sequence in the Banach space E, and let A be a positive real number. Then P{ supllXnll ? A}

sup E [IIXo11]

This maximal inequality is a good illustration of the usefulness of condition (B). Another illustration of its usefulness is the next stopping result.

5.2. Martingales and amarts

191

(5.2.20) Definition. Let (Xf)fEN be a stochastic process with values in a Banach space E, and let C be a subset of E. We say that the process (Xn) stops outside C if X,, (w) C implies X,, (w) = Xt,+1(w). (5.2.21) Definition.

Let (Xn)nEiN be a process adapted to the stoSuppose (Xf)fEN has values in a Banach space E, and let C be a Borel subset of E. The first entrance time of (Xn) in E \ C is the (possibly infinite) stopping time or defined as follows. Let D = {Xn E C for all n}. Then chastic basis

for w E D .

The process (Xn) stopped outside C is the process (Yn) defined by Yn = XnAa-

(5.2.22) Proposition. Let C be a closed bounded subset of the Banach space E. Suppose the process (Xn)nEIN in E satisfies condition (B). Let (Yn) be the process (Xn) stopped outside C. Then

E

[sup IIY.II] < 00. n

Proof. Let D and a be as in the definition. Outside the set D, we have convergence Yn -* Xo, so by Fatou's lemma, E [IIXoII 1S2\D] no.

Fix m > no. Choose D E Fm so that IIE [Xm 1D] II >- AE p IIE [Xm 1A] II - E.

Then choose n > m so that IE [Xn 1sz\D] II < e.

Define TEEby

m T =

{ n

on D

on St \ D.

Now IIE [Xr] II no and A E .F,, choose m > a and let m

on A

Q

onIl\A,

or

so that a' E E and a' > no. Then IIE[Xo1A]II N,

P{I(Xn-X,x*)I >E} <S.

Banach-valued random variables

200

Write An = { I (Xn - X, x*) I > e}. Then

E [I (Xn - X, x*)I] < E [I(X,, - X, x*)I 1A,.] + E [I(X- - X, x*)I 10\A, ]

E} < let=E. r

Now Yn -> Y a.s., so there is m1 > mo with P{IIYn

- YII>E forsome n>ml}<E.

206

Banach-valued random variables

Then, with the triangle inequality, we have P{ IIXn - EF°° [X] II > 3e for some n > ml } < 3e. Since c was arbitrary, Xn -> E,"°° [X] a.s. For proof of convergence in Bochner norm, observe that E-F^ [X] is uniformly integrable. Here is the observation that will be used to strengthen the scalar convergence results proved above to strong convergence results. If a martingale (or even a uniform amart) converges in a very weak sense, then in fact it converges in a much stronger sense.

(5.3.21) Proposition. Let (Xn)nE N be a uniform amart in E. (1) Suppose (Xn) is L1-bounded. Then Xn converges scalarly if and only if Xn converges a.s.

(2) Suppose (Xn) is L,,-bounded. Then Xn converges scalarly if and only if Xn converges in the Bochner norm. (3) Suppose (Xn) satisfies E [supra IIXnII] < oo. Then Xn converges scalarly if and only if Xn converges in the Bochner norm.

Proof. First observe that it is enough in all parts to prove the result in the case that (Xn) is a martingale by the Riesz decomposition (5.2.13). Observe that a.s. convergence implies scalar convergence. Also, for a martingale (Xn), Bochner norm convergence (to Xc,.) implies mean convergence E [(Xn, x*) - (X., x*)] -> 0; these are scalar-valued martingales, so they converge a.s.; thus Xn converges scalarly to X,,.. (2) follows trivially from (3). We begin with (3). Let (Xn) be a martingale with

E

[sup IIXnII] < oo. n

Suppose Xn converges scalarly to X. Now if A E 1:'n,, for some m, then by the dominated convergence theorem, we have

lim E [((Xn, x*) - (X, x*)) 1A] = 0 n-oo for all x* E E*, so that E [Xn 1A] --> E [X 1A] weakly. But for n > m the sequence E [Xn 1A] is constant. Thus E [X 1A] = E [X,,, 1A]. This is true for all A E Jl;n, so that E.F [X] = Xm Then by the preceding result, Xn converges a.s. and in Bochner norm to X. For part (1), we will use the same stopping technique as before. Fix A > 0 and let Yn be obtained by stopping Xn outside

CA={xEE:IIxII 0 in probability, i.e.

limsupP{IIH"11>e}=0

oEE T>O TEE

a.s. for every e > 0. Millet & Sucheston [1980b] introduced pramarts and proved that real-valued pramarts converge a.s. if they satisfy condition (d): liminf IJX; 111 + liminf IIX; 111 < oo.

For pramarts, this condition is strictly weaker than Ll-boundedness (but not for martingales or amarts). Pramarts have the optional sampling property, and therefore there is a continuous parameter theory (see Frangos [1985]). Millet & Sucheston [1980b], Egghe [1981], Slaby[1982] and Frangos [1985] considered almost sure convergence of Banach valued pramarts, and

Banach lattice valued "subpramarts." The Banach valued case is included in the result of Talagrand stated below. The process (Xn) is a martingale in the limit if Hmn --+ 0 a.s., that is, lim sup IIHmnll = 0 a.s.

m-oo n>m

216

Banach-valued random variables

Every pramart is a martingale in the limit. This definition is due to Mucci [1973] in the scalar-valued case. Bellow & Dvoretzky (unpublished) and Edgar [1979a] proved partial results on the a.s. convergence of martingales in the limit. These results were also improved by the result of Talagrand, which is stated below. The process (Xn) is a Talagrand mil if Hon --+ 0 in probability, that is, for every e > 0 there exists N such that for all bounded stopping times or > N and all integers n > v, we have P{IjHnll > e} < E. Every martingale in the limit (and therefore every pramart) is a Talagrand mil. Talagrand [1985] showed that (if E has the Radon-Nikodym property) all Ll-bounded Talagrand mils converge a.s. A large class of processes is the game which becomes fairer with time (GFT). The process (Xn) is a GFT if Hmn -+ 0 in probability, that is, for

every e>0there is Nso that if n>m>N,then P{IIHmnil >El <e. The definition is due to Blake [1970] in the scalar case. This very general class contains all those mentioned above. Mucci [1973] and Subramanian [1973] proved independently (in the scalar case) that a uniformly integrable

GFT converges in mean. The vector case can be proved from a Riesz decomposition as follows.

(5.3.40) Theorem. Let E be a Banach space with the Radon-Nikodym property. Let (Xn)nEIN be a uniformly integrable GFT. Then Xn converges in Bochner norm.

Proof. Write Hmn = Efm [Xn] - Xn,, for m < n. Since

,,,,n -> 0 in

probability, and this double sequence is uniformly integrable, we get Hmn -->

0 in Bochner norm. Thus, for a fixed m, the sequence (Em [Xn])n°__",, is Cauchy in Bochner norm. Therefore it converges in Bochner norm. Write

Y,n for the limit, and Z n = X,n - Y,,,.. Thus we have Xn = Yn + Zn, where (Yn) is a uniformly integrable martingale, and (Zn) converges to 0 in Bochner norm. By the Radon-Nikodym property, (Yn) converges in Bochner norm.

(5.3.41) (Failure of strong convergence of amarts.) Let E = 12 and let

{eni:nEIN,1 2(Y(t)+Z(t)) 0. But then b(Z)) > cfi(X)} > 0, so

(5.4.4b)

E [2 (V)(Y) + (Z)) I > E [,O(X)] + 2-m

for some m E IN. We may define Y, = Eon [Y] and Zn = Eon [Z] to obtain martingales (Yn) and (Zn) in C with Xn = (Yn + Zn). But Y. converges 2 to Y and Zn converges to Z, so by the continuity (and boundedness) of (5.4.4b) implies that there is no > m such that for all n > no,

E

12

('(Yn) + tP(Zn))] > E [,G(Xn)] + 2-m.

Banach-valued random variables

222

By (5.4.4a), we have

El

P(X.) 2! 2 (V '(Y') + V (Z')) - V (Xn) - 2-n ,

so E [VJ(Xn+1) - V(Xn)] > 2-1-2-n. But Xn converges, and ip is bounded and continuous, so by the dominated convergence theorem

E [V(Xn+l) - b(Xn)] -+ 0. This is a contradiction. Common fixed points for noncommuting maps The Schauder fixed-point theorem asserts that a weakly continuous map of a weakly compact convex set C into itself has a fixed point. The MarkovKakutani fixed-point theorem asserts that a commuting family of continuous affine maps of a weakly compact convex set C into itself has a common fixed point. The Ryll-Nardzewski fixed-point theorem is a generalization

to certain noncommuting families of maps. A martingale proof of a key element of the theorem is given here.

(5.4.5) Definition. Let C be a convex set. A map S: C -+ C is affine

if

n

S

n

ti xi i-1

ti Sxi

= i=1

forxiEC,ti>0,Eti=1. (5.4.6) Definition. A family S of maps from C to itself is distal if for

any pair x,yECwith x0y,wehave infSESIISx-SyII>0. (5.4.7) Theorem. Let C be a closed bounded convex Radon-Nikodym set in a Banach space E. Let S1, S2, be (not necessarily continuous) affine maps from C into itself. Suppose that {S1, S2,. } generates a distal 2-iSi. Then semigroup. Let xo E C be a fixed point for the map S = xo is also a fixed point for each of the maps Si.

Proof. Let (Un) be an independent sequence of random variables with val-

ues in the countable set IS,, S2,. } and P{Un = Si} = 2-i for all i. Note that E [Unx] = Sx for x E C. Let .fin be the v-algebra generated by U1, U2,.

Un. Define X.: 11 --+Cby Xn(W) = Ul(W)U2(W) ... Un(W)x0

We claim that (X,,) is a martingale. (In the following calculations, measurability is clear, since the v-algebras .Fn are atomic.) E''F" [Xn+1] = EF" [U1U2 ... U,Un+lxo] = U1U2 ... [Un+1x0] = U1U2 ... UUSxo = U1U2 ... UnxO = Xn.

5.4. Geometric properties

223

Suppose Sixo 0 xo for some i. There is 6 > 0 so that I I SSixo - Sxo I I > 6 for

all S in the semigroup generated by {S1, S2, 2-i and on the set {U.+1 = Si}, we have

}. Now P{U,,+1 = Si}

IIX.+1-XnII=IIU1U2...U.(Sixo-x0)II?6,

so P{II Xn+1- X. II ? 6} > 2-i. By Egorov's theorem, the martingale (X,a) does not converge a.s. This contradicts the fact that C is a Radon-Nikodym set. The conclusion can be extended to include uncountable sets of maps by the usual abstract considerations. We will outline them briefly.

(5.4.8) Corollary. Let E be a Banach space, and let C C_ E be closed, bounded, and convex. Assume: (1) C has the affine fixed-point property; that is, any weakly continuous affine map of C into itself has a fixed point.

(2) C has Corson's property (C); that is, a family of closed convex subsets of C with the finite intersection property has nonempty intersection. (3) C is a Radon-Nikodym set. Then any distal semigroup S of weakly continuous affine maps from C to itself has a common fixed point.

Proof. For each S E S, let F(S) x E C : Sx = x }. By the affine fixedpoint property, each F(S) is nonempty. Clearly, each F(S) is closed and convex. Now C is a Radon-Nikodym set, so by Theorem (5.4.7), the family { F(S) : S E S } has the countable intersection property. Therefore, by property (C), the family has a nonempty intersection, that is, S has a common fixed point.

Finally, we can use a single-map fixed-point theorem to deduce the RyllNardzewski fixed-point theorem itself.

(5.4.9) Theorem. Let C be a weakly compact convex set, and let S be a distal semigroup of weakly continuous affine maps from C into itself. Then S has a common fixed point. Proof. Weakly compact sets have property (C) [since closed convex sets are weakly closed] and are Radon-Nikodym sets (5.3.33). The fixed point property is a version of the Schauder fixed-point theorem (Dunford & Schwartz [1958], V.10.5). Thus, Corollary (5.4.8) applies to this situation. Dentability In the geometric arguments below we will use the following notations. Let E be a Banach space. The closed ball about a point x with radius r is denoted B(x,r)={yE E: IIy - xII < r}.

224

Banach-valued random variables

Let C C E be a nonempty bounded set. A slice of C is a set of the form S(C, x*, a) = { y E C : x*(y) > sup x*(C) - a } for some x* E E* and some a > 0. Here sup x* (C) = sup { x* (y) : y E C } by definition. Dentability is a purely geometric concept that turns out to be related to the Radon-Nikodym property and to martingale convergence.

(5.4.10) Definition. Let C be a nonempty closed bounded subset of a Banach space E. We say that C is dentable if for every e > 0, there is a point xo E C such that xo 0 cl conv (C \ B(xo, e)).

The Hahn-Banach separation theorem shows how slices are related to convexity. This proof is given in great detail. Future uses of the HahnBanach theorem will be less verbose.

(5.4.11) Proposition. Let E be a Banach space, and let C C_ E be a closed, bounded, and nonempty set. Then C is dentable if and only if C admits slices of arbitrarily small diameter.

Figure (5.4.11) A slice of a dentable set. Proof. Suppose C is dentable. Let e > 0 be given. Then there is xo E C so

that xo 0 cl conv (C \ B(xo, e)). By the Hahn-Banach separation theorem, there is x* E E* and y E IR such that sup x* (C \ B(xo, e)) < y < x*(xo) (see Figure (5.4.11)). But then

{yEC:x*(y)>y} is a slice of C with diameter less than 2e.

5.4. Geometric properties

225

Conversely, suppose C has slices of arbitrarily small diameter. Let e > 0 be given. Then there is a slice S(C, x*, 2a) of C with diameter less than E. Let xo be any point in this slice. Then (C \ B(xo, c)) fl S(C, x*, a) = 0,

so that (C \ B(xo, E)) C { y E C : x* (y) < sup x* (C) - a } . So also

cl conv (C \ B(xo, E)) C {y E C: x* (y) < sup x* (C) -a } , and xo is outside the closed convex hull.

It will be useful to observe that a nonempty bounded set C is dentable if its closed convex hull is dentable, since every slice of cl conv C contains a slice of C. The next theorem is probably the most often used geometric characterization of the Radon-Nikodym property.

(5.4.12) Theorem. Let E be a Banach space, and C C_ E a nonempty closed bounded convex set. Then C is a Radon-Nikodym set if and only if every nonempty closed convex subset of C is dentable.

Proof. Suppose that there is a nonempty closed bounded set D C C that is not dentable. Then cl conv D C C is not dentable. So we may assume D is convex. We will construct a quasimartingale on [0, 1) with values in D C_

C that diverges everywhere, yet is very close to a martingale and should converge a.s. by Lemma (5.4.3) together with (5.3.34). (Alternatively, the uniform amart convergence theorem (5.3.22) could be applied, since every quasimartingale is a uniform amart.)

Since D is not dentable, there is e > 0 so that every slice of D has diameter greater than E. Thus if x E D, then D fl B(x, E/2) contains no slice of D. This means that cl conv (D \ B(x, e/2)) = D.

Let Q = [0, 1), P = Lebesgue measure, and F = Borel sets. A process on (Q, .F, P) will be constructed recursively. (The construction is

similar to that of the martingale constructed in (5.4.4).) Choose xo E D (this is possible since D is nonempty), let F1 = { 0, 9 }, and define X1 (w) = xo. Now suppose X1, X2, .. , X,ti have been defined, and X9, is a simple function measurable with respect to a a-algebra .F,, with atoms [0, t1), [t1, t2), ... , [tN-1, 1) -

Banach-valued random variables

226

Let x be one of the values of Xn; say X, (w) = x for w E [tj_1,t3). Now

D = cl conv (D \ B(x, a/2)), so there exist x1i x2, , xk E D with JJx - xi 11 > 6/2 for all i, and scalars a1, a2, , ak > 0 with Ek1 ai = 1 such that k

x-aixi

< 2-'t.

i=1

Subdivide the interval [tj_1i tj) into disjoint subintervals with lengths proportional to the ai, and define Xn+1 on these intervals to have the values xi. Thus: IIXn(w) X.+1(w)II > 2 for all w E fl

-

and IIXn

- E.cn [Xn+1] II 0 we will write the average range of µ on M this way: µ(W)

a(M) =

P(W)

W EF,W CM,P(W)>0

.

We will prove first:

for every s > 0, and every M E F with

P(M) > 0, there is F E.F with F C M, P(F) > 0, and diam(a(F)) < s.

(5.4.12a)

Now a(M) C C, so it is dentable. Thus there is a slice

S={yEa(M):x*(y)>ry} with S # 0 and diam S < e. Now x* o p is a scalar measure and x* o p ^y} fl M. Then P(F) > 0 since S 54 0. And a(F) C_ S: Indeed, for W C F with P(W) > 0 we have µ(W)/P(W) E a(M) and *

X

µ(W) P(W)

E [Z 1w]

E ['Y lw] P(W) > P(W) = 7

so µ(W)/P(W) E S. This completes the proof of (5.4.12a).

5.4. Geometric properties

227

Now, take a maximal pairwise disjoint collection {Mi } in F such that diam a(Mi) < e. We have now proved for every e > O,there is a countable partition

(5.4.12b)

{ {M} of Il with diam a(M)<e for all i. }1)ono-1

Thus, we can choose a sequence ({MM of partitions, increasing with respect to refinement, such that diama(M$) < 2-ri-1. Define n

Xn -

00

P(Mzn) 1M: . a

i=1

Now if M E F, we have IE[Xn1M-p(M)]

=E <E

i

p(M,) 1M^nM p(M:)

[2-n1MflM]

p(MM f1 M) 1M^nM

P(mi n m)

;

< 2-n-1.

i

Thus E [Xn 1M] -+ A(M). Now the estimates we have will be used to show that the process (Xn)nE]N converges uniformly. If w E Q and n E IN, then w E Min for some i and w E MJn+1 or some j. But

p(E a

(Mn) Xn(w) = p(Mn)

(Mn),

and, since Mn +1 C M, , p(Mn+1)

Xn+1(w) = p(Mn+1) E a(Mi ). Therefore IIXn(w) -Xn+1(w)II < 2-n-1. Let Y be the uniform limit of the sequence Xn. Then, for any M E.F we have both E [Xn 1M] -> p(M) and

E[Xn1M]->E[Y1M],sop(M)=E[Y1M]. (5.4.13) Theorem. Let E be a Banach space. Then E has the RadonNikodym property if and only if every nonempty closed bounded convex subset of E is dentable. Proof. Combine Theorem (5.4.12) with Proposition (5.3.3). Strongly exposed points There is a more precise characterization of Radon-Nikodym sets in terms of strongly exposed points.

228

Banach-valued random variables

(5.4.14) Definition. Let C be a nonempty closed bounded convex set. A point xo E C is a strongly exposed point of C if there is x* E E* (the strongly exposing functional) such that x* (xo) = sup x* (C) and if xn E C is a sequence with limn-oo x*(xn) = x*(xo), then limn-o. II xn - xoll = 0.

Clearly, a strongly exposed point of C is an extreme point of C. If C has a strongly exposed point, then C is dentable. The geometric criterion we will prove is due to Phelps and Bourgain: a set C is a Radon-Nikodym set if and only if every nonempty closed bounded convex subset of C is the closed convex hull of its strongly exposed points. To begin the proof, we have a lemma, which we prove using Lemma (5.4.3). Recall the notation for slices:

S(C, x*, a) = { y E C : x* (y) > supx* (C) - a }.

(5.4.15) Lemma. Let E be a Banach space, and let C C_ E be a nonempty closed bounded convex Radon-Nikodym set. Let x* E E* with IIx* II = 1, let E > 0, and 0 < rl < a. Write M = sup x* (C). Then there is xo E S(C, x*, ii) such that

xo V cl conv [(C\ B(xo, E)) U { x : IIx - xoll S 1, x*(x) < M - a } ] .

x*=M- rl x*=M Figure (5.4.15) Illustration for Lemma (5.4.15).

Proof. Let W = (C\B(xo,E))U{x: IIx-xoll < 1, x*(x) <M-a}. (The shaded portion in Figure (5.4.15).) Assume the conclusion is false. Let

xo E C be such that x*(xo) > M - 71. Let 6 = x*(xo) - M + q > 0. Then xo E cl conv W, so xo is as close as we like to a convex combination of elements x1 i x2, , Xk E W, so for each xi, either Ilxo - xi II > E or both llxo - xi II < 1 and x* (xi) < M - a. If the second case holds, then

a - ij < x*(xo) - x*(xi) < llx*II llxo - xill. Hence for all xi we have Ilxi - xo 11 ? min{E, a - 77).

5.4. Geometric properties

229

The above observation may be used to construct a bounded process (Xn) on S2 = [0, 1] that stops when it leaves C and satisfies: (1) Xi(w) = xo a.s.; (2) IIE1 [Xn+,] - X. II < S 2-n;

(3) if x*(Xn(w)) > M - r7, then we have Xn(w) E C and either IIXn+1(w) - Xn(w)II > e or both x*(Xn+1(w)) < M - a and II Xn(w) - Xn+1(w)II < 1;

(4) if x*(Xn(w)) < M - 77, then Xn+1(w) = Xn(w). Now if x*(Xn(w)) > M - 77, then IIXn+i(w) - Xn(w)II > min{E,a - 77}. But C has the Radon-Nikodym property, so Xn(w) converges a.s. by (5.4.3). Write X00(w) for the limit, so that x*(X00(w)) < M -,q a.s. So E [x*(X00)] < M - 77. But IIE [X00]

-

00 xo11

n-i

IIE [Xn+1] - E [Xn]

E,

so that M -,q > x* (E [X.]) > x*(xo) - S > M - q, a contradiction. The preceding lemma can be used to prove a geometric consequence.

(5.4.16) Lemma. Let E be a Banach space, let C C_ E be a nonempty closed bounded convex Radon-Nikodym set, and let 0 > 0. Then the set

As = {x* E E* : 11x* 11 = 1 and diamS(C,x*,a) 0} is dense in the unit sphere of E*. Proof. Let x* E E* with 11x* 11 = 1, and let b > 0. We will show that there is y* E AO with IIx* - y* II < S. We may assume that C is contained in the unit ball and is not just a single point. We may assume that x* is not

constant on C, since functionals not constant on C are dense in the unit sphere of E*. Let c > 0 be so small that

2e M-a. But that means x*(x) > M- a- x*(xo) > M- a- M= -a.

Similarly, using xo - x in place of xo + x, we see that x*(x) < a. Thus Ix*(x)l < a for all x E E with IIxii 1 and y*(x) = 0. So by Proposition (5.4.1), either Ilx* + y*II < 2a or 11x* - y*Il < 2a. Since 2a = 4e < 6, it remains only to show 11x* + y* II > 4e.

Suppose (for purposes of contradiction) that iIx* +y*II < 4e. Now we have l le < sup x* (C) - inf x* (C) and M = sup x* (C), so there is x1 E C with x*(x1) < M - 11e. Then

y*(xi) = -x*(xi) + (x* +y*)(xi) ? -x*(x1) - IIx* +11*II

> -M+11e-4e=-M+7e. But sup y* (C) < y* (xo) + e, hence y*(xi) < y*(xo) +'F = -x*(xo) + (x* + y*)(xo) + e

0 satisfy supy* (W) + 2e < M, and choose x* E A with Ilx* - y* 11 < F. Say x* strongly exposes xo E C. Then y*(xo) > x*(xo) - e

and x*(xo) > M - e, so y*(xo) > M - 2e > supy*(W). Thus xo 0 W, a contradiction.

There exists a converse of the preceding result. Suppose, for each nonempty closed convex subset D of C, the set D is the closed convex hull of its strongly exposed points. Then C is a Radon-Nikodym set. In fact, by (5.4.12), it is enough if each nonempty convex D C C has at least one strongly exposed point, since a set with a strongly exposed point is necessarily dentable.

5.4. Geometric properties

231

Complements

(5.4.18) (Locally convex space.) Suppose that F is a locally convex Hausdorff topological vector space. Let K C_ F be a nonempty compact convex metrizable subset. Then there is a linear subspace FO C F containing K, a Banach space E, and a continuous linear transformation T : FO -> E such that T is a homeomorphism on K. (5.4.19) (Choquet's theorem.) The preceding can be used to prove Choquet's theorem in one of its original forms: Suppose that F is a locally convex topological vector space. Let K C F be a nonempty compact convex metrizable subset. If xo E K, then there is a random variable X : !Z --+ K

such that P{X E ex K} = 1 and the Pettis integral E [X] exists and is equal to xo. (5.4.20) (An application of Choquet's theorem.) A function f : 7L -> C is called positive definite if 00

E tjtkf(j - k) > 0

j,k=-oo

for any choice of complex numbers {tj},'?__.O with all but finitely many equal to 0. We will outline a use of Choquet's theorem to prove the following

theorem of Herglotz: If f : 2Z -> C is positive definite, then there exists a finite measure p on [0, 21r) such that 2ir

f(n) =

L

for all n E 7L. The following steps are used for the proof. (a) Using sequences {tj}'?_ with only one nonzero term, it can be

seen that f (O) > 0, and using sequences with only two nonzero terms, it can be seen that If (n) I < f (0) for all n. (b) The set K = { f E l,,,, (7L) : f positive definite, f (0) = 1 } is a compact convex metrizable subset of the locally convex space l,,.(7L) with its weak* topology. (c) For 0 E [0, 27r) let go(n) =

einB

Then gg E ex K.

(d) Any extreme point f of K is of the form go. This can be seen as follows: Let a = Re f (1), and show that f ± g E K, where g(n) = 4 f (n + 1) - 2 -a f (n) + 4 f (n - 1),

so that g = 0. Let /3 = Im f (1). Then f f h E K, where h(n) = 4i f (n + 1) - 2 of (n)

- 4i f (n - 1),

so that h = 0. Finally, f (n + 1) = f (1) f (n) for all n, so that f has the form gg. (This proof is similar to a proof in Edgar [1983].)

232

Banach-valued random variables

(5.4.21) (Krein-Milman property.) We say that a Banach space E has the Krein-Milman property if every closed bounded convex subset is the closed convex hull of its extreme points. (Or, equivalently, every nonempty closed bounded convex subset has at least one extreme point; see Bourgin [1983], Proposition 3.1.1.) Comparing this to Theorem (5.4.17), one might naturally conjecture that the Krein-Milman property is equivalent to the Radon-Nikodym property. This is still an open problem. Remarks

The selection theorem (5.4.2) is due independently to Yankov [1941] and von Neumann [1949].

Choquet's theorem (for compact sets) is due to Choquet [1956]. The proof by two-point dilations follows Loomis [1975]. The generalization to Radon-Nikodym sets is due to Edgar [1975]. For further reading on Choquet's theorem, we recommend Phelps [1966]. Ryll-Nardzewski [1967] gave a proof for his fixed-point theorem using a "differentiation" argument. Namioka & Asplund [1967] realized that a condition like dentability could be used in the proof. The use of property (C) in our proof may be new. Property (C) holds, for example, in separable Banach spaces, in weakly compactly generated Banach spaces, and in many others. See Corson [1961], Pol [1980]. The Radon-Nikodym type of argument used here to prove the fixed-point theorem will also prove this variant (Namioka & Phelps [1975], Theorem 15): If C is a separable weak-star closed bounded convex subset of the dual F* of a Banach space F, then any distal semigroup of weak-star continuous affine maps of C into itself admits a common fixed point. The geometric characterization (5.4.13) was one of the first results showing that the Radon-Nikodym property is relevant in the study of the geometry of Banach spaces. First Rieffel [1968] showed that the Radon-Nikodym property follows from dentability of every subset. (This step was simplified by Girardi & Uhl [1990].) Then the converse was proved in small steps by Maynard [1973], Davis & Phelps [1974], Huff [1974]. The geometric characterization (5.4.17) is due to Phelps [1974] and Bourgain [1977]; the use of martingales for the proof was suggested by Kunen & Rosenthal [1982].

5.5. Operator ideals This section shows a few more of the connections between our subject matter and the geometric theory of Banach spaces. More knowledge of Banach space theory is required for an understanding of this section than was required in the previous sections.

One concept that has been much used in recent years in the study of Banach spaces is the "operator ideal." We will consider here primarily the following operator ideals: the absolutely summing operators, the RadonNikodym operators, the Asplund operators. Absolutely summing operators The first ideal that we will consider here, because of its connections with amarts, is the ideal of absolutely summing operators.

5.5. Operator ideals

233

(5.5.1) Definition. Let E be a Banach space, and let {xi}°_1 be a sequence of vectors in E. We say that the series E xi is convergent if there is a vector y such that n

lim

y - E xi

n-*00

=0.

i=1

The series > xi is absolutely convergent if 00

Ilxill 1. (1) For any pair E, F of Banach spaces, the set Hp (E, F) of all pabsolutely summing operators from E to F is a Banach space under the norm 1rp.

(2) For any linear transformation T, we have IITI1 < irp(T), with equality for rank one operators. In particular, if T is p-absolutely summing, then T is bounded. (3) If T : E -+ F is p-absolutely summing, Q : El -+ E and R : F -+ F1

are bounded operators, then the composition RTQ: El -+ Fl is p-absolutely summing, and irp(RTQ) 5 IIRII irp(T) IIQII.

5.5. Operator ideals

235

Proof. (2) Take n = 1 in the definition. If x E E, then IITxII < irp(T) sup { I (x, x*) I : x* E E*, IIx* II

1

Therefore IITxII < irp(T) IITII So we have IITII < irp(T).

If T : E -+ F has rank 1, then

T (x) = xo (x) yo

for some xo E E* and yo E F, Ilxoll = 1. Thus IITII = Ilyoll, and for xl, x2i

, xn, E E, we have n

IITxiIIp =

Jxo(xi)Ipllyollp

i=1

< IITll sup { (E l x* (xi) I P) : x* E E*, IIx* II < 1 }

.

Thus 7rp(T) < IITII in this case, so 7rp(T) = IITII. (1) The only nontrivial assertion is the completeness. Suppose (T,,,) is a sequence of p-absolutely summing operators, Cauchy in the norm 1rp. Then irp(T,n) converges, say to a. By part (2), the sequence (T,n) is also Cauchy in the operator norm. So there is an operator T : E -+ F with II Tn, - T11 -+ 0. We claim that T,,, -+ T also in the norm irp. Now T,nx -+ Tx for all x E E. If the finite set {x1, x2, , xn} is given, then we have

In

n

1/p

1/P

= hm

IITxiIIp i=1

IlTmxillp i=1

n

< lim irp(T,n) sup m

IIx*II:51

= asup

(t1(Xi'X*)JP i=1 1/p

I (xi,x*)IP i=1

So T is p-absolutely summing. Given e > 0, we may choose mo so that +7rp(T,,,o - Tm) < e for all m > mo. The argument just used shows that 7rp (T,no - T) < e. So we have proved that irp (T.,,,, - T) --+ 0. (3) follows from the definition.

One of the most useful results on absolutely summing operators is:

(5.5.5)

Pietsch factorization. Let p > 1, and let E, F be Banach

spaces. Write K for the unit ball of E* with its weak* topology. Then

the operator T : E -+ F is p-absolutely summing if and only if there is a probability measure y on K and a constant C such that

/f

1/p Ix*(x)Ipdµ(x*)\)

IITxII < C I JK

Moreover, the smallest such constant C is 7rp(T).

for all x E E.

Banach-valued random variables

236

Proof. Suppose that T : E -p F is p-absolutely summing, and irr (T) = 1. Then IITII < 1. For X E E, define gx E C(K) by gx(x*) = Consider two subsets of C(K) defined by

Ix*(x)IP.

F1 = { f E C(K) : sup f (x*) < 1 I x* EK l

F2=conv{gx:xEE, IITxhI=1}. Then F1 and F2 are convex sets and F1 is open. Since irr(T) = 1, we have F1 fl F2 = 0. So the two sets can be separated by a linear functional on C(K). By the Riesz representation theorem, there is a positive constant A and a signed measure p on K with variation 1 so that f E F1 implies

f E F2 implies

JK

fK

Since F1 contains all nonpositive functions, the measure µ is a positive measure. Since F1 contains the open unit ball of C(K), we have A > 1. Thus for any x E E with IITxII = 1 we have fK Ix*(x)IPdp(x*) > 1, and for every x E E we have fK Ix* (x) IP dp(x*) > IITxIIP. For the converse, suppose that p and C exist. Then if xl, x2i , xn are in E, we have n

/ IITxihIP < CP J

lx*(x=)IPdl-p(x*)

K

i=1

F be an operator. Then T is absolutely summing if and only if there is a constant C such that, for every random variable X in E, we have E [IITXII] E is a random variable, we have by Fubini's theorem E [IITX II ] < 7rl (T) E [1K 1x* (X) I dµ(x*)]

JE [Ix*(X)I] dµ(x*)

_ 7r1(T)

K

< 71 (T) sup E [Ix*(X)I] ir1(T)IIXIIp Conversely, suppose there is a constant C so that

IITXII1 0. Let .TT be the a-algebra generated by {X1, X2, , Xn}. The process (Xn) is adapted to the stochastic basis In fact, (Xn) is an amart potential.

Indeed, given e > 0, there is N such that 00

1: anxn

<e

IIfl=N

for any choice of scalars an with lanl < 1. If or E E and or > N, write

Dn=Cnfl{Q=n}, so that

cc

E [Xe] =

E P(C) P(Dn)

But P(Dn)/P(Cn) < 1, so IIE [Xe] II < e. Thus (Xn) is an amart potential. Next, we must show that (TXn) is not bounded on any set of positive measure. Since > P(Cn) = oo, the Borel-Cantelli lemma tells us that for almost all w E St, we have w E Cn for infinitely many n. For those n, IITxnII

IITXn(w)II = P(Cn)

_

1

Tn

But Nn -* 0, so the sequence TXn(w) is unbounded.

As a corollary we obtain some interesting characterizations of finitedimensional spaces.

(5.5.11) Theorem. (1) A Banach space is finite-dimensional if and only if every amart in E is a uniform amart. (2) A Banach space E is finite-dimensional if and only if every L1bounded amart in E converges strongly a.s. (3) A Banach space E is finite-dimensional if and only if every L1bounded amart in E converges weakly a.s. (4) A Banach space E is finite-dimensional if and only if every L1bounded amart potential in E is bounded on a set of positive measure.

Proof. In finite dimensional spaces, the convergence properties follow from

the one-dimensional cases: the projection on a one-dimensional space is continuous, and a sequence in 1R' converges if and only if each of its coordinates converges. The converse directions follow from the previous theorem, together with Theorem (5.5.8).

5.5. Operator ideals

241

Radon-Nikodym operators The next operator ideal to be considered is the ideal of Radon-Nikodym operators.

(5.5.12) Definition. Let E and F be Banach spaces, and let T : E -> F be a bounded linear operator. We say that T is a Radon-Nikodym operator (or T has the Radon-Nikodym property) if, for every probability space (S2, , P) and every vector measure µ:.F --> E that is absolutely continuous with respect to P and has finite variation, the measure Tp has a RadonNikodym derivative in L1(S2, .F, P; F). This is motivated by the definition of the Radon-Nikodym property for a Banach space E. In fact, the Banach space E has the Radon-Nikodym

property if and only if the identity operator on E is a Radon-Nikodym operator. Most of the elementary part of the theory of Banach spaces with the Radon-Nikodym property can be reproduced in terms of RadonNikodym operators, simply by inserting the operator into the proofs in appropriate places. For example, as in (5.3.3), it is sufficient to use vector measures µ with bounded average range in E. Or: an operator is a RadonNikodym operator if and only if its restriction to each separable subspace is a Radon-Nikodym operator. The next result states that the collection of Radon-Nikodym operators is a Banach operator ideal. The "Radon-Nikodym norm" of T is simply the operator norm IITII.

(5.5.13) Proposition. (1) For any pair E, F of Banach spaces, the set of all Radon-Nikodym operators T : E -+ F is a Banach space under the operator norm IITII.

(2) If T : E - F is a Radon-Nikodym operator, Q : El --* E and R: F - F1 are bounded operators, then the composition RTQ: El - Fl is also a Radon-Nikodym operator. Proof. (1) To see that the set of all Radon-Nikodym operators from E to F is a vector space, note that

d(aTp) dTp =a dP dP d(Ti +T2)µ _ dTip dT2,a dP dP + dP Next consider completeness. Suppose operators Tn : E - F are RadonNikodym operators, and Tn converges to T in the operator norm. By taking a subsequence, we may assume 00

I IITn+1 - TnhI < 00n=1

Banach-valued random variables

242

Now each difference T7L+1 -Tn is a Radon-Nikodym operator, so there exist random variables Xn : 0 -+ F so that Xn = d(T,,+1 - Tn) p/dP. (Let To = 0.) Now the variation satisfies I (Tn+1 - T.),1 (9) X,,, converges in Ll (1; F), say to X. For A E .F, E [X 1A] = limo E

[(x)

1AJ

m T,,, (µ(A)) = T (µ(A))

Thus X = dT u/dP, so T is a Radon-Nikodym operator.

(2) Let (1,.F, P) be a probability space, and let µ:.F -+ El be a vector measure, absolutely continuous with respect to P, with finite variation. Then QM: F --> E is a vector measure, absolutely continuous with respect to P, with variation at most IIQII IAI(1) < oo. Thus the Radon-Nikodym

derivative X = dTQµ/dP exists. But then the Radon-Nikodym derivative RX = dRTQp/dP exists. Thus, the composition RTQ is a RadonNikodym operator. Arguments almost identical to those used above (5.3.29 and 5.4.12) prove the next result.

(5.5.14) Proposition. Let E and F be Banach spaces, and let T : E -p F be a bounded linear operator. Then the following are equivalent: (1) T has the Radon-Nikodym property; (2) every closed bounded nonempty set C C_ E has slices S with image T(S) of arbitrarily small diameter in F; (3) for every L1-bounded martingale (Xn)nEIN in E, the image (TX,,,) converges a.s. in F; (4) for every amart (Xf)fEIN in E satisfying condition (B), the image (TX,,,) converges scalarly in F.

The Radon-Nikodym operators are related to the Riesz representable operators. Before we make this more precise, let us discuss the representable operators.

(5.5.15) Definition. Let (1, .F, P) be a probability space, and let E be a Banach space. Then an operator T : L1(St, .F, P) -> E is said to be representable if there is a random variable X : Q --+ E such that T(Z) = E [Z X] (Necessarily X E L,,,, (1l,.F, P; E).)

for all Z E Ll (St, .F, P).

5.5. Operator ideals

243

(5.5.16) Proposition. Let E be a Banach space. Then E has the RadonNikodym property if and only if every operator T : L1 -+ E is representable.

Proof. There is a one-to-one correspondence between the set of all operators T: L1(Il, F, P) -+ E and the set of all absolutely continuous vector measures u: F -+ E with bounded average range. The measure p corresponding to the operator T is defined by p(A) = T(1A). The operator T is representable by a random variable X if and only if the measure p has Radon-Nikodym derivative X.

Analogous (roughly speaking) to the Pietsch factorization is the LewisStegall factorization, which is proved next.

(5.5.17) Theorem. Let (St, F, P) be a probability space, and let E be a Banach space. Then the operator T : L1(fl, F, P) -+ E is representable if and only if it factors through the space 11, that is, there exist operators S: L1 -+ 11 and R: 11 -->E such that T= RS. S

L1

11 I

IR

E The range of a representable operator T : L1(Il,

P) -> E is separable.

Proof. Suppose first that T has a factorization T = RS. Then S is representable, since ll has the Radon-Nikodym property (5.1.16). Say S(Z) = E [Z X] for all Z E L1. But R is continuous and linear, so we have RS(Z) = E [Z R(X)], so that RS is also representable. Conversely, suppose that T is representable. Then there exists X E L,,. (S2, .F, P; E) so that T (Z) = E [Z X] for all Z E L1. Now X has separable range, so T also has separable range. We may assume that E is a separable space. Let e > 0. For each positive integer n, the space E can be covered by countably many balls of radius E2_n-1, so there is a random variable Yn : Il -> E with countably many values such that

IIX-Ynll,,. <E2-n-1. Let X1 =Yl andXn=Yn-Yn_1 forn> 2, so that Xn has countably many values and IIX - E; =1 X n ll < e2-n-1. Then Xn has the form Xn = E00xnk 1Enk, k=1

where Xnk E E and Enk E F with Enk fl Enk' = 0 (for k # k') and II xnk Il < 3E2-n+l if n > 2. Define S : L, (Q, F, P) S(Z) (n, k) = Ilxnkll E [Z 1Enk]

11(IN x IN) by

Banach-valued random variables

244

for Z E L1. Then we have 00

00

IIS(Z)II11 = i i IIXnkII jE [Z 1E,.k] n=1 k=1 00

00

00

< E IIx1k11 E [IZI lE,k] + E E 3E2-n+1E [IZI 1E,.k] n=2k=1

k=1

But IIX

- X, 11 < E/4 and IIXIIo0 = IITII, so IIX1k1I < IITII + E/4. Then IIS(Z)II1 s (IITII + 4E [IZI] + 4 E [IZI] _ (IITII + e) E [IZI]

Thus IISII

.

IITII + e. Then define R: 11(IN x IN) -+ E by 00

00

R(h) _

h(n, k) n=1k=1

IlxXnk

nk II'

with the convention 0/0 = 0. Then IIR(h)II E I h(n, k) I = IIhII, so IIRII < 1. Also 00

00

RS(Z) = E

Xnk

IlxnkII E [Z1E,.k]

IIXnkII

n=1 k=1 00

E n=1

1Enk (fxflk k=1

Z

00

=EE[XnZ] n=1

=E[XZ] = T(Z). Thus T has been factored, and IIRII IISII

IITII +E.

If T is a representable operator, then it factors through 11. Since 11 is separable, the range of T must be separable as well.

The connections between the representable operators and the RadonNikodym operators are illustrated by the next two results.

(5.5.18) Proposition. Let (S2, F, P) be a probability space, and let E be a Banach space. Then the operator T : L1(S2, F, P) -p E is a RadonNikodym operator if and only if it is representable.

Proof. Suppose that T is a Radon-Nikodym operator. Define p: F -+ Ll by µ(A) = 1A. Then p is absolutely continuous with respect to P, and

5.5. Operator ideals

245

jp1(A) = P(A) for all A E F, so that p has average range bounded by 1. Since T is a Radon-Nikodym operator, there exists a Radon-Nikodym derivative X = dTµ/dP. Now if Z = 1A, then we have E [Z X] = E [X 1A] = Tp(A) = T(1A) = T(Z). The equation E [Z X] = T(Z) holds for simple functions Z, and each side is a continuous linear function of Z E L1, so the equation is true for all Z E L1. Therefore T is representable. Conversely, suppose T is representable. Then T factors through 11; say

T = RS, where S: L1 -+ ll and R: 11 -+ E. If I is the identity operator on 11, then I is a Radon-Nikodym operator, since the space ll has the Radon-Nikodym property (5.1.16).

I

Therefore, by the ideal property (5.3.13(2)), T = RIS is a Radon-Nikodym operator.

(5.5.19) Theorem. Suppose E and F are Banach spaces, and suppose T : E -+ F is a bounded linear operator. Then T is a Radon-Nikodym operator if and only if, for every probability space (S2, .F, P) and every

operator S: L1P) -+ E, the composition TS is representable. Proof. If T is a Radon-Nikodym operator, then so is TS, and therefore TS is representable. Conversely, suppose TS is representable for any S. Then if p: .F -+ E is a vector measure with bounded average range, there is a unique operator S: L1(1

P) --+ E

such that S(1A) = u(A) for all A E.F. But TS is representable, say by X : 1 -+ F, and then X = dTµ/dP. In light of the Lewis-Stegall factorization, this result can be restated like

this: T : E -+ F is a Radon-Nikodym operator if and only if, for every operator S : L1(1k, .F, P) -+ E, there is a factorization of TS through 11: L1

>

11

sI 1

Banach-valued random variables

246

It is not hard to see that there is a relation between the ideal of RadonNikodym operators and the ideals of compact and weakly compact operators: An operator T : E -* F is compact if the closure of the image of the unit ball of E T(BE) is a compact set. Equivalently, for every bounded sequence (x,,,) in E, there

is a subsequence (x,,,,,) such that T (x,,,,) converges in F. The operator T : E -+ F is weakly compact if the closure of the image of the unit ball of E is a weakly compact set. Equivalently, for every bounded sequence (x,,) in E, there is a subsequence (x,,,,,) such that T (x,,,,) converges weakly in F.

(5.5.20)

Proposition. Every weakly compact operator is a Radon-

Nikodym operator.

Proof. Let T : E -> F be weakly compact. Let (Xf)fEIN be a martingale in the unit ball of E. Then the martingale (TX,,) has values in a weakly compact convex set, namely the closure of the image of that unit ball under T. Now weakly compact convex sets are Radon-Nikodym sets (5.3.33). So (TX,,) converges. This shows that T is a Radon-Nikodym operator.

(5.5.21) Corollary. Every compact operator is a Radon-Nikodym operator. A weakly compact operator T : Ll -+ X has separable range and is representable.

Proof. Any compact operator is weakly compact. Suppose T : Ll (St, F, P) -+ E is weakly compact. Then T is a RadonNikodym operator, so it is representable. Thus there is a Lewis-Stegall

factorization T = RS, where S: Ll -+ ll and R: ll -> E. But ll is separable, and the range of T is a subset of the range of R, so it is separable. Asplund operators

The next operator ideal to be considered is the ideal of Asplund operators. The main result of concern here is the connection between scalar convergence and weak almost sure convergence.

(5.5.22) Definition. Let E and F be Banach spaces, and let T : E -+ F be an operator. Then T is an Asplund operator if T* is a Radon-Nikodym operator. When defined in this way the ideal properties follow easily from those of the Radon-Nikodym operators.

(5.5.23) Proposition. (1) For any pair E, F of Banach spaces, the set of all Asplund operators T: E -+ F is a Banach space under the operator norm 1IT11.

(2) If T : E -+ F is an Asplund operator, Q : El -> E and R: F -> Fl are bounded operators, then the composition RTQ: El -+ FL is also an Asplund operator.

5.5. Operator ideals

247

As with the ideals considered above, this one has close connections with

factorization conditions. In order to properly state these conditions, we first consider the Haar operator. The Cantor set is the countable product topological space

A={0,1}IN. It is compact and metrizable. It is made up of made up of parts

IAni:nEIN,O IR are defined by hni = 10n+1,2i - 10n+1,2i+1

If { eni : n E IN, 0 < i < 2n } is an enumeration of the unit vector basis of 11, then the Haar operator H : l1 -+ L,,.(A, p) is defined by

H(eni) = hni

For the Here is Stegall's theorem characterizing Asplund operators. proof, see Stegall [1981], p. 515, or Bourgin [1983], Chapter 5. A function cp : E -+ IR is Frechet differentiable at the point x E E if there is a linear functional x* E E* such that lim Icc(x + y) - cP(x) - x*(y)I = 0. I1yII-0

II.II

248

Banach-valued random variables

(5.5.25) Theorem. Let U: E -* F be an operator. Then the following are equivalent. (1) U is an Asplund operator. (2) U* factors through a space with the Radon-Nikodym property. (3) U* maps closed bounded convex subsets of F* to Radon-Nikodym subsets of E*. (4) If cp: F -> IR, is continuous and convex, then coU is F7rechet differentiable on a dense subset of E.

(5) U is not a factor of the Haar operator H:

EUF 11 )H L,,. (6) U factors through a space, every separable subspace of which has separable dual.

A Banach space where the identity operator is an Asplund operator is called an Asplund space. Then E is an Asplund space if and only if E* has the Radon-Nikodym property. The interest here of Asplund operators is the following result.

(5.5.26) Theorem. Let E and F be Banach spaces, and let U: E -> F be an operator. The following are equivalent. (a) U is an Asplund operator. (b) If XX,, is a sequence of E-valued Bochner measurable functions, converging scalarly to 0, with sup,, IIXnII < oo a.s., then the sequence (UXn,) almost surely converges weakly to 0. (c) If (Xn.)n,EIN is a weak sequential amart potential in E satisfying condition (B), then (UXn) almost surely converges weakly to 0. (d) If (Xf)n,EIN is an amart potential in E satisfying condition (B), then (UXn,) almost surely converges weakly to 0.

Proof. Three of the four parts of the proof are easy. (a) (b). Suppose U is an Asplund operator. Then by Theorem (5.5.25(6)), U can be factored, U = WV, where V : E - Eo, W : Eo -> F, and Eo is a Banach space, every separable subspace of which has separable dual. Let Xn : St -* E be Bochner measurable, sup,, IIXn Il < oo a.s., and suppose X,, converges scalarly to 0. Then by (5.1.7) there is a separable

subspace El of Eo such that P{VX,, E E1} = 1 for all n. Now El is separable; choose a countable dense set {x* }0 For each k there is a null set 1 k C_ SZ such that (VXn,x*) -* 0 on St \ Qk. Now since {xk} is dense and {jjVXn(w)jj} is bounded, we have (VXn,x*) -+ 0 for all x* E El and all w E SZ \ U Qk. But then UXn = WVXn converges weakly to 0 a.s.

5.5. Operator ideals

249

(b) (c). Let (Xf)fEIN be a weak sequential potential in E satisfying condition (B). If x* E E*, then (X,,, x*) is a scalar amart potential, and thus (X,,, x*) -+ 0 a.e. That is, (X,,) converges scalarly to 0. By the maximal inequality (5.2.19), sup,, IIXnIl < oo a.s. Therefore, UXn -> 0

weakly a.s.

(d). Any amart potential is a weak sequential potential. (a). Suppose U is not an Asplund operator. By Theorem (5.5.25(5)), U is a factor of the Haar operator H: (c) (d)

EUF ST

IT

) L. (A,µ)

11

H

Now we claim that it will suffice to exhibit an amart potential (Xf)fEJN in 11 satisfying condition (B) such that HX,, does not converge weakly a.s.

to 0 in L,,. Indeed, then (SX,,) will be a strong amart potential in E satisfying condition (B), but (USX,,) will not converge weakly to 0 a.s. in F. For this example, we let 1 = A be the Cantor set, and P = µ the natural measure on A. We will also use the other notation in Definition (5.5.24). Define X,,: A --> ll by n 2m-1 Xn(w) =

hmi(w)emj.

m

We will prove that (Xn) has the required properties. First, observe that (Xn) is Lam-bounded, and hence satisfies condition (B), since for any w E St, n 2m-1

IIXn(w)Illl

=

EE m=1 i=0

n

n

1

hmi (w)

We claim next that HXn does not converge weakly a.s. to 0 in Since the range of H is contained in the subspace C(A), it suffices to show

that HXn does not converge weakly to 0 in C(0). For each w E A, we have

n 2m-1

HXn(w)(w) = n

n

hmi(w) hmi(w) = n E 1 = 1, m=1 i=0

so that HXn(w) does not converge weakly to 0.

m=1

Banach-valued random variables

250

Now we claim that if 2 < p < oo, there is a constant CP such that for any scalars ani with IaniI < 1,

e En 2--1

(5.5.26a)

< CP nP/2.

ami hmi

m=1 i=0

This is a classical inequality. It is implicit from Paley [1932]. A modern proof might observe that the Haar functions, taken in the proper order, constitute a martingale difference sequence. The square function S associated with (5.5.26a) satisfies S2

n 2"`-1

n

=

1 = n,

ami h- n.i
1; for example, take p = 7, a = 1/4. Let en = n-°`. Given x* E li with 1Ix*11 < 1, write ani = x*(eni), so that land < 1. By (5.5.26a), P

l

nn` 2TM-1

E Y ami n m=1 i=0

CP

< np/2

hmi P

and hence for A > 0, 1

P{

n 2m_1

> n1 m=1 i=0 ami hmi

CP np/2AP

LetaEE,v>N. Then 00

1

2"`-1

n

E [Ix*XI] = t E n- E n=N

= n=N"" 00

LI

m=1 i=0

°O

1

ami hmi 1{o=n} 2m -l

P

n m=1 i=0 ami hmi 1{o=n} > A

=n (E"Pf...JdA+EPf...IdA)

1

dA

5.5. Operator ideals

251

00

E (rnP{v=n}+npP n=N

En

00

00

_E

n-a P{a = n} +

n=N

N-aP{o' = n} + p C

=N-a+

Cp1

00

E n(p/2)-a(p-1)

°O

n =N

P

C

i E n=N nPl2en (p - 1)

00


oo, since a > 0 and p/2 - a(p - 1) > 1. Now it is easy to verify that (Xn) is an amart: IIE [Xo] II

= sup

I (E [XoJ , x*) I

IIx* II J send the canonical unit vectors in 11 to the sequence en in J. Then T is not weakly compact, since en has no weakly convergent subsequence. If (Xn) is a weak amart in 11, then it is in fact a strong

amart (by the Schur property; see Dunford & Schwartz [1958], IV.8.14). But then Xn converges scalarly, so TXn converges scalarly, and (since J is an Asplund space) TXn converges weakly a.s. (5.5.34) (Incompleteness of the Pettis norm.) Let E be a Banach space. Then L' (St, F, P; E) is complete under the Pettis norm II 11p if and only if E is finite-dimensional. To see this, apply Theorem (5.5.9) to the identity operator T : E -> E. If 11 . 11p is complete, then by the closed graph theorem, X --> TX is a bounded operator in both directions between 11 11p and II - 11L, -

Remarks

The brief treatment of p-absolutely summing operators, the Pietsch factorization, and the Dvoretzky-Rogers lemma follows Lindenstrauss & Tzafriri [1979]. For further reading on operator ideals and their applications to Banach space theory, see Pisier [1986]. Bellow [1976a] proved that Ll-bounded vector amarts converge strongly only if the Banach space is finite-dimensional (Theorem (5.5.11(2))). Edgar & Sucheston [1977a] proved that Ll-bounded vector amarts converge weakly only if the Banach space is finite-dimensional (Theorem (5.5.11(3))). Ghoussoub [1979b] showed how absolutely summing operators come into the problem. The operator ideal of Radon-Nikodym operators was introduced by Reinov [1975] and Linde [1976], who independently developed its important properties. The Lewis-Stegall factorization is from Lewis & Stegall [1973].

Stegall [1981] discusses the ideal of Asplund operators, but the name is from Edgar [1980], which contains Theorem (5.5.26).

The characterization of weak amart convergence (5.5.28) is due to Brunel & Sucheston [1976b]. Theorem (5.5.27) (the dual is separable if and only if scalar convergence implies weak convergence) is due to Brunel & Sucheston [1977].

6

Martingales

In this chapter we will look at martingales more carefully. In many cases,

submartingales and supermartingales will be included. We will consider some of the many interesting results known for these processes, and some of the applications of them. We discuss many well known results concerning maximal inequalities, La, laws of large numbers for martingale differences (extended to a class of mixing processes), decompositions, convergence of transforms of martingales, Burkholder's square-function inequalities. The Maharam lifting theorem is derived from the martingale theorem.

6.1. Maximal inequalities for supermartingales In this section we will discuss various maximal inequalities for martingales and supermartingales. We will also prove a law of large numbers for martingale differences. As an application of this, we present a proof of the law of large numbers under the condition of star-mixing. Let (S2, a', µ) be a a-finite measure space, let (.F11)11EIN be a stochastic

basis, and write F,,, = a(U.Fi). We will write Ei for the conditional expectation: if X is a measurable function,

This may or may not exist: see (2.3.8). We will also write Pi, for the corresponding "conditional probability" : If A is a measurable set,

Pi(A) = Ei 1A. In particular, El will play the role usually reserved for the expectation E. A reader not interested in this generality may assume that µ is a probability measure, and El = E. However, the.F1 variant has applications: see (6.1.3) below.

A maximal inequality

Let A > 0 be a random variable, measurable with respect to F1. Let (Xi) be an adapted sequence of random variables. Fix a value of n E IN.'

For1 0 a.e. and that E1Xi = EF1 [Xi] exists for all i. It follows that EjX1 exists for all j, and that EjXX exists for a E E. Now if we apply El to both sides of (6.1.1a), we obtain: n-1

(6. 1.1b) AP1

sup Xi 11 0 for all n > no. Assume (Xn) is eventually positive. When we let n --> oo in (6.1.1b), we obtain AP,

sup Xi > A } 1

JJJ

Now a conditional probability is bounded by 1, so we obtain:

Martingales

256

(6.1.2) Proposition. If (Xn) is an eventually positive supermartingale, then for each positive .F1-measurable random variable A, we have

P1fi>l supXz>A

(6.1.2a)

minIA X1,1}.

_I

JJJ

(6.1.3) Corollary. Let (Xn) be an eventually positive supermartingale, let r E INT be fixed, and let A be an Fr-measurable random variable. If A < supz>1 Xi a.s., then A < supl 0 be an F1-measurable random variable. Then for each n,

\P1

(6.1.4a)

1 X, >A sup t EEXX+1 Xi and also EiX +1 > 0, so EiX +1 > X, . Therefore we have (EzX +1 - XZ)/ai+1 > 0. Now in order to apply (6.1.1b) to Y2=X2 Then Xi >A 1 0. There is p E IN such that I Ei `-j (xi/ci) I < e

for p < j < m. Note that m

m

m

xi

E xi = 1:(Cj - cj-1) E Ci-, i=j

j=1

i=1

Therefore if p < m we have 1

-CmE xi i=1

E(Cj - cj_1)

E xi-

Cm

j=1

i=j

-CE (cj - cj-1)

m

P

M

m

M

j=P+1

i=j

1

Ci

m

xi Ci

P m < 1- E(Cj - cj_1) 2xi- +Cm te Cm Cm

i=j Ci

j=1

Now when m --> oo, the first term goes to 0, so we obtain

<e,

lim sup m-1oo

which proves the existence of the required limit.

(6.1.7)

Proposition. Let p > 1 be a fixed number. Let (X,,,) be a

nonnegative submartingale, and let (ai) be a positive increasing predictable sequence with ai -+ oo. Suppose 00

XiP+1 p-

El

(6.1.7a) ti=1

a,+1

\\\

X P I< oo. /

Then limn_00 X,,,/a, = 0 a.e. Proof. First, by Kronecker's lemma (6.1.6) applied to (6.1.7a), we have r

(6.1.7b)

El

P am+1

r

1

El

m

1 1

am+1

XiP

El

= i=1

L

a m+1

0. J

By Jensen's inequality (2.3.10), the process (XP,) is also a submartingale. , Xx), Let m < n; apply (6.1.4a) to the finite submartingale (X n, X n+1)

6.1. Maximal inequalities for supermartingales

259

replacing ai by ail and A by AP; apply the conditional expectation E1; then take the limit as n -+ oo. The result is:

l <El X-P rr >AP7 APP1Ssup XP P P + 1111:>. ai

a

11J

El rX +1P - XiP

00

i=m

L

J.

ai+1

The first term on the right converges to 0 by (6.1.7b). The second term converges to 0 by (6.1.7a). Therefore limX,, /a,, = 0 a.e. The following theorem is due to Paul Levy if p = 2. The particular case where the Yi are independent and with zero expectation is one of the two classical Kolmogorov laws of large numbers. (The second one is proved below: see the case p = 1 of (6.1.15).)

(6.1.8) Theorem (Y. S. Chow). Let p be a fixed number, 1 < p< 2. Let M,, = E 1 Yi be a martingale, and let ai be a positive increasing predictable sequence. Suppose cc

E1 CafO 0. Now G'(x) = pH(x)/xP+1 where H(x) = 2 - (1 + x)P-1 - (1 - x)p-1, so we must show H(x) > 0 for

0 < x < 1. But H(0) = 0 and H'(x) = (p -1) ((1- x)P-2 - (1 +x)P-2) > 0 since p - 2 < 0 and 1 + x > 1- x. This completes the proof of the inequality (6.1.8d). Applying (6.1.8d) to (6.1.8c), we obtain

Ei(IMM+1I1 - Mill < 2Ei(IMMIP+ IYi+1IP) -2E2(jM2IP)

= 2Ei(IYi+1IP). Then apply El to obtain (6.1.8b).

Much more difficult is the case p > 2. The proof is based on a fundamental inequality of D. L. Burkholder, proved below (Theorem (6.3.6)). For simplicity we will take ai = i. The exponent in the denominator must be 1 + p/2 < p, so the condition is more difficult to satisfy than the corresponding condition with exponent p. The independent case of the following is due to Brunk [1948] and Chung [1947, 1951].

(6.1.9) Theorem (Y. S. Chow 1967b). Let p be a fixed number with 2 < p < oo. Let Mn = > 1 Yi be a martingale such that 0 l

ff

(6.1.9a)

E1

i1+p/2 +P/2

< 00.

i=1

Then limMn/n = 0 a.e. Proof. By Holder's inequality (with exponents p/(p - 2) and p/2), we have

:

n

i=1

2/p

n

2 < n1-2/P

IYIP i=1

6.1. Maximal inequalities for supermartingales

261

Now raise both sides to the power p/2 and apply E1: P/2

n

E y2

< nP/2-1E1 EIYIP i=1

i=1

Next we apply the right side of (6.3.6) with Sn = (E a constant K < oo such that

(

n

(6.1.9b)

E1

Yi

1

Y2)1/2. There is

p)

1 be a family of a-algebras; typically An is generated by a single random variable Y n . Let . P n = a(Uii=1 bi 18;, where bi > 0 and Bi E A,,,,+n Choose A E C

such that P(A) > 0. Then

IE[ZkIA]-E[Zk]I < i=1

b P(BZn P(A)A) P(BinA) P(A)

a

bi

- i=1 b2P(Bi) P(Bi)

i=1 J

f (n) > biP(Bi) = f (n)E [Zk]. i=1

Then if k -> oo, we obtain IE [X+ I A] - E [X+] I < f (n)E [X+].

This remains true if A is replaced by C (integrate over sets A E C), and also if X+ is replaced by X-. The lemma follows.

If (Yn) is a stochastic process, we will say it is star-mixing if the aalgebras An = a(Yn) are star-mixing.

6.1. Maximal inequalities for supermartingales

263

(6.1.11) Theorem. Let (Y,,,) be star-mixing, mean 0 (E [Y,,,] = 0), and L1-bounded (IIY,, II 1 < K < oo). Let p be such that 1 < p < oo. Assume that (6.1.11a) E001 E [I YYI P] /iP < oo, if p < 2, or (6.1.11b) E001 E [IYYIP] /i1+P/2 < oo, if p > 2. Then

1 n

(6.1.11c)

Yi --* 0

a.s.

i=1

Proof. Fix an integer k with 0 < k < N, and apply Lemma (6.1.10) to the process X,,,, = YmN+k If W. = Q(YmN+k, 1'(m-1)N+k, m > 1 and WO = {S2, 0}, then IEnm-1

[YmN+k] - E [YmN+k] I < f (N)E [I YmN+k I]

i

1'N+k) for

.

Now E [YmN+k] = 0 and E [IYmN+kI ] < K, so given e we may choose N so

large that (6.1.11d)

IExm-1 [YmN+k] I

1 and 0 < k < N. Fix kandN. LetT,,,=YmN+kform> 1. Then U,,, = T,,, - Exm-1 [T,,,] defines a martingale difference sequence. The sequence (T,,,.) satisfies (6.1.11a) or (6.1.11b), as appropriate. Now "centering," that is replacing by U,,,, does not change this, because IUmI < ITmI +Exm-1 [ITmI] A/2} + P{S" > A/2}

< -E[M1]+-E[R1] Next, if X is a martingale, then by (6.2.2) sup,, E [IX,,1] = E [M1] + E [R1], which proves (i). If X is a positive submartingale, then E [M1] _

limp T E [Xp] = sup,, E [X,,] and R1 = M1 - X1 implies that E [R1] < sup,, E [X,,], which proves (ii).

Part (i) is proved by Burkholder [1973] with constant 3 rather than 6. Martingale transforms Let X = (Xn) be a sequence of random variables. Its difference sequence

is the process Y = (Yn) so that Xn = 1 Y. If V = (Vn) is another process, then the transform of X by V is the process Z = (Zn) given by Zn = 1 ViY. We will sometimes write Z = V * X. If Xn is the fortune of a gambler at time n, then the transform Zn may

be viewed as the result of controlling X by V. If we assume that V is predictable, then multiplication of Y1 by V is equivalent to changing the stakes for the ith game on the basis of information available before the ith game.

If X is a martingale, and V is predictable, then the transform Z = V * X is also a martingale (see Section 3.2). We will now prove that the transform of an Ll-bounded martingale by a bounded predictable process converges. This is a surprising result, since (see (6.2.8)) the transform need not be Ll-bounded.

(6.2.6)

Theorem. Let X be an Li-bounded submartingale or an L1-

bounded supermartingale. If V is an Lam-bounded predictable process, then the transform Z = V * X converges a.s. Proof. We may assume that IV,, I < 1 for all n by multiplying by a constant.

If X is an Ll-bounded submartingale, and X = X- X" is the Krickeberg decomposition (6.2.2), then V * X = V * X' - V * X", and X', X" are both positive supermartingales; therefore it is enough to prove the result when X is a positive supermartingale.

272

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First suppose the positive supermartingale X is Lam-bounded, say Xn < c

a.s. Let X = M - A be the Doob decomposition of X. Since V * X = V * M - V * A, it suffices to show that both of these parts converge a.s. Convergence of V * A follows from the monotone convergence of An to Ate, since (V * A)n - (V * A).-11:5 I V.1 I An. - An-1I

0 Now note that E [X3_lYj] = E [EFi-1 [Xj_lYj]] = E (or = 0 in the martingale case). Hence applying the expectation, we have

IISn-1II2 + II`Kn-1II2 < 2E [Xn-1Xn]

with equality in the martingale case. Now by the optional sampling theorem (1.4.29), the process (X,-An) is also a martingale (or positive submartingale). So this inequality remains true: IISrn(n-1)I12 + IIXrn(n-1)I12 < 2E [Xrn(n-1)Xrnn]

6.3. Norm of the square function

275

with equality in the martingale case. Now we have I XTnn I < I XT I except on the set {T = oo}, where I XTAnI < A. Hence the sequence (I XTA(n_1)XTnnI ) is dominated by the integrable random variable A(I XT I + A). Now XTnn ->

XT as n -+ oo, even on {T = oo}, by the convergence theorem (1.2.5). So we obtain the first inequality of (6.3.1a) with the dominated convergence theorem. We now prove the submartingale analog of Theorem (6.1.3). Recall that this theorem was proved for positive supermartingales (Lemma (6.2.4)) using Krickeberg's decomposition. The estimates given here for positive submartingales are sharper than those in Proposition (6.2.5(ii)).

(6.3.2) Lemma. Let X be a martingale or a positive submartingale; let X* be its maximal function and S its square function. Then for every A > 0 (6.3.2a)

P{X* < A, S > A} < 2 IIXIIi,

(6.3.2b)

P{S > A} < a IIXIIi.

Proof. By Proposition (6.1.5), we have AP{X* > A} < IIXII1, so (6.3.2a) implies (6.3.2b). So it suffices to prove (6.3.2a). Let r = inf { n : I Xn I > A }. On the set A = {T = oo}, we have X* < A. Hence by Lemma (6.3.1),

P{X* < A, S > Al < P{ST > Al < A2[ST] E 3A} < 3E [Xn

Proof. Let Z. = Xn 1{s, (ex)>a} Let T = inf { n : Sn (8X) > A }. On the set B = {Sn(8X) > 3A, Xn < A}, we have Sn(8X) > A since 0 > 1, hence

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276

T A holds on B. Now Z is a submartingale, since the product of an increasing adapted sequence and a positive submartingale is increasing. Applying is a submartingale and the process Lemma (6.3.2) to the submartingale ,Zn,Zn,...),

(Z1,Z2,...

we obtain AP{Xn < 13A, Sn(OX) > Al < AP{Zn

A, Sn(Z) > A}

2 IIZIII = 2E [Xn

Now applying the classical weak Doob's inequality (1.4.18), we have AP{Xn > Al < E [Xn 1{x;,>A}] < E [Xn 1{ n>A}]

.

Combining,

AP{Tn > /3A} < AP{Xn > Al + AP{Sn(BX) >,3A, Xn < Al < 3E [Xn 1{n>A}]

.

Now that we have a weak inequality, we may use the three-function inequality to pass to a strong inequality between the process X and its square function S.

(6.3.4) Proposition. Let X = (Xn) be a positive submartingale. Let

1 Xn7 Xno ..')-

In fact the proof of the decomposition is easy in the present case; the general case is only invoked to motivate the choice of M and R: Set

Mn=limT P

{XP] =XZ,

Mj = hm T E-'j [XP] = E-'j [Xn ]

for 1 < j < n - 1.

Now if Rj = Mj - Xj, then M = (Mj) and R = (Rj) are both positive martingales, with X = M - R. Note that Mn = Xn and Rn = Xn ; so IiMnllp A}]

(6.3.7a)

for constants a,,3. Then, if c is the constant in the (02) condition, i.e. e(l3u) < c(u) for all u, and d > 0, then

T

0, and Q = 1 + 282. Then

30(e 11Sn.4

1) IIXnII.P.

Proof. Apply the preceding Theorem, using Lemma (6.3.3), where a = 3.

(6.3.9) (Square function and maximal function.) Since the control of the square function S by X (6.3.8) is similar to control of the maximal function

X* by X (3.1.13), one may expect that S may be equivalent to X*. This is indeed the case: Let -P be an Orlicz function such that 4)(u)/u -+ oo, satisfying the (A2)

condition 4'(2u), < c4(u) for some constant c > 0. Then there exist constants b, and Bc such that for every martingale X, we have bcMt [S(X)] BcM4[S(X)] (Burkholder, Davis, & Gundy [1972]). An L1 version of this result is also true: There exists constants b, B such that for every L1-bounded martingale X, we have bE [S(X)] < E [X*] < BE [S(X)] (B. Davis [1969]). There is further a recent extension to more abstract spaces due to Johnson & Schechtman [1988]. (6.3.10) (Converse false.) Corollary (6.3.8) shows that if sup IIXn11,p < oo, then sup IISn Ikk < oo. The converse is false: choose Xn = X1 not in L logk L;

then Sn = 0 is in L logk-1 L. (6.3.11) (Best constants.) The inequalities in (6.3.6) can be improved.

Let 1 < p < oo, 1/p + 1/q = 1, and p* = max{p, q}. Then the constant c, can be replaced by (p* - 1)-1, and Cp by p* - 1. The left-hand side inequality is optimal if 1 < p < 2; the right-hand side if 2 < p < oo. The best constants in other cases are not known. The results extend to martingales taking values in a Hilbert space. (See D. L. Burkholder [1991], Theorem 3.3, or Burkholder [1988].) Remarks

The first version of the inequality (6.3.6) is in Burkholder [1966].

6.4. Lifting This section will discuss liftings for measure spaces. A measure defines an equivalence relation on the measurable sets: two sets are equivalent if they agree almost everywhere. A lifting is a choice of one set from each equivalence class in such a way as to preserve as much of the structure as possible. (The technical definition is given below.) The existence of liftings may be established using the martingale convergence theorem, so the topic is appropriate for discussion here. Liftings have also been used to choose a good basis for differentiation of integrals.

6.4. Lifting

281

Throughout Section 6.4, we will write (S2, F,,u) for a nonzero complete v-finite measure space.

We will write A - B to mean u(A L B) = 0. Similarly, we will write f - g to mean { w E fl : f (w) 54 g(w) } has measure 0.

In some places, it will be important to distinguish between an equivalence class of measurable functions and an actual function. In order to do this, we will use the following notational conventions. For 0 < p < ao,

the notation Lp (SZ, )',A) = Lr(u) = £ will denote the set of all realvalued .-measurable functions f on S1 with f If I" dp < oo. The notation L,, (St, F, µ) will denote the set of all bounded real-valued F-measurable functions on 9. The notation Lr (fl, will denote the quotient of LT, (S2, F, µ) obtained by identifying functions that agree almost everywhere.

If f E LP, the corresponding equivalence class will be written with a dot, f', when it is important to distinguish between the two. For f E Lam, we will write II f I I for the uniform norm:

IIfII..=sup{If(w)I:wEQ}, and

I f II

for the usual L,,,, seminorm:

IIfII,,,, =inf{MEIR:p{w:If(w)I>M}=0}. We also write IIf'II. = Ilf II. for the quotient norm on L. Existence of liftings

There are two common formulations of the basic definitions concerning liftings. We use here the two terms lifting and density to distinguish between them. (Some authors use "lifting" to mean what is here called "density," and others use "density" for what is here called "lower density.") Roughly speaking, a "lifting" chooses one element of each equivalence class of L,,, in a nice way.

(6.4.1) Definition. A lifting for the measure space (fl, F, p) is a function

p: L (S2, , p) -> L. (Sl,,F, µ) satisfying (i) P(f) = f; (ii) if f g, then p(f) = p(g); (iii) p(1) = 1;

(iv) if f > 0, then p(f) > 0; (v) p(f + g) = p(f) + p(g); p(af) = ap(f), a E 1R; (vi) P(f g) = P(f)P(g) (In fact, condition (iv) follows from the others.)

(6.4.2) Definition.

A density for (fl,

satisfying (a) 6(A) = A;

(b) if A = B, then b(A) = b(B); (c) 6(fl) = Il; 6(0) = 0; (d) b(A n B) = b(A) n b(B); (e) b(A U B) = b(A) U b(B).

p) is a function 6: F -- F

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282

A lower density is a function 6 satisfying (a), (b), (c), (d). An upper density is a function 6 satisfying (a), (b), (c), (e). If g is an algebra of subsets of fZ that meets each µ-equivalence class of F exactly once, then g is the range of a unique density, namely 6(A) is the element of g in the equivalence class of A. Conversely, if 6 is a density for (S2,

then the range of 6 is an algebra of sets that meets each

equivalence class exactly once. If 6 is a lower density, then the function 6' defined by 6'(A) = S2 \ 6(11 \ A)

is an upper density, and 6'(A) 6(A). We will say that 6' is the upper density corresponding to 6. If 6 is a density, then 6' = 6. It will take some effort to produce a nontrivial example of a lifting or a density. But upper and lower densities can be easily exhibited. Consider the measure space IR with Lebesgue measure A. The Lebesgue density theorem (7.1.12) can be used to show that 6(A)

tl x : lim

o

A(A n (x - h, x + h)) = 2h

1

11

defines a lower density (called Lebesgue lower density),

6'(A)=

x:limsupA(An(x2h ,x+h))

hj0

>0

defines an upper density, and 6(A) C 6'(A). Liftings and densities are closely connected.

(6.4.3) Proposition. (1) Let p be a lifting. If A E F, then the function p(1A) is an indicator function. The equation p(1A) = 16(A) defines a density 6.

(2) If 6 is a density, then there is a unique lifting p such that p(1A) _ 16(A) for A E F.

Proof. (1) Let p be a lifting. If A E F, then by condition (vi) of Definition (6.4.1), we have p(1A)2 = p(1A) = p(1A). So for all w E S2, either p(1A)P) = 0 or p(1A)P) = 1. Therefore, p(1A) is an indicator function, say 16(A). Properties (a), (b), (c) of Definition (6.4.2) for 6 follow from properties (i), (ii), (iii) of Definition (6.4.1) for p. Next, note

that p(1O\A) = p(1 - 1A) = 1 - p(1A), so 6(11 \ A) = Il \ 6(A). Also 16(AnB) = P(1AnB) = P(1 A 1B) = P(1A)P(1B) = 16(A) 16(B) = 16(A)n6(B),

so 6(AnB) = 6(A)n6(B). Finally, by complementation, we get 6(AUB) _ 6(A) U 6(B).

(2) Write S for the set of F-simple functions. If f = ci 1A: E S, define p(f) = E ci,16(Ai). It can be checked that p is well defined and IIP(f)II.. = 11f 11,,.. The conditions (i) - (vi) hold for functions in S. The closure of S in the G,,.-norm is G,,. (12, F, µ), sop can be extended uniquely to 4, in such a way that IIP(f)IIu = Iif III holds for all f E Gam. Now it can

6.4. Lifting

283

be easily checked that (i) - (vi) hold for functions in G,. Suppose that p' is another lifting for p) that satisfies p'(1A) = 16(A) for A E F. Clearly

p(f) = p' (f) for f E S. But if 11f 11x) = M, then -M < f < M almost everywhere, so -M < p(f) < M everywhere, and thus IIp'(f)IIv. < 11f 11".. Thus p and p' are both continuous, so they coincide on Gam. This proposition shows that the problem of existence of a lifting is equiv-

alent to the problem of existence of a density. The next few results have the goal of proving such existence. The first one has a standard sort of Zorn's lemma proof, but checking the details takes some space. (6.4.4) Proposition. Let S be a lower density for the nonzero complete orfinite measure space (52, -F, µ), and let b' be the corresponding upper density. Then there is a density 6* for (52, F, p) satisfying b(A) < 6*(A) < 6'(A) for

allAE.F. Proof. Consider the collection R of all 9 satisfying: is an algebra of subsets of 52, meets each p-equivalence class at most once,

6(A)9AC6'(A)forall AE9. Then R is partially ordered by inclusion, and not empty, since {0, 52} E R.

The union of a chain of elements of R is again an element. By Zorn's lemma, R has a maximal element g*. It remains only to show that 9* meets each equivalence class at least once. So let AO c -.F, and define

Ao= U (6(CUAo)\C). CEcc*

We will show that Ao Ao and Ao E G*. First, if C, D E G*, then

6(CuAo)n6(DU(52\Ao))\(CUD)CS(CuD)\(CUD)=0, so (6(C U Ao) \ C) n (6(D U (Sl \ AO)) \ D) = 0. Therefore, if D E G*, we have

Aon (6(Du(1 \A0))\D) =0 by the definition of A', we get 6(Ao) C A'. By completeness, we have A' E F and A' A0. Next, if D E

we have

6((Q\D)u(52\A'0))=b((I \D)u(1\Ao)) c (1 \D)U(52\A'); so D n Ao C S'(D n A'). Now if E E 9*, then

6((l\E)uA'0)=6((l\E)uAo) c (1 \E)uAo, soE\A' C6'(E\A').

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284

Let G' be the algebra generated by G* and A', so

G'={CU(DnA')U(E\A'):C,D,EEG*}. and we claim G'ER. IfBEG',sayB=CU(DnA')U

Then (E \ A'0), then (since 6' is an upper density), B C 6'(C) U 6'(D n A'0) U 6'(E \ A'0) = b'(C U (D n A'0) U (E \ A'0)) = 6'(B). Also Q \ B E G', so 1l\B C_ 6'(Il\B) or 6(B) C_ B. To show that G' meets each equivalence class at most once, suppose B1, B2 E G' and B1 - B2. Then u(Bl A B2) = 0, so

B1AB2 C8'(BIAB2)=O, soB1=B2Therefore ' E R. By the maximality of g*, we have G' = G*. Therefore Ao E G*. This completes the proof that G* meets each equivalence class. Now g* is the range of the required density 6*.

(6.4.5) Corollary. Lebesgue measure on the real line IR admits a lifting. Proof. Apply Proposition (6.4.4) to the Lebesgue lower density. Note that if (1, F, p) is a complete measure space, and if G C F is a a-algebra containing all of the y-null sets in F, then (St, G, pjG) is also a complete measure space. The collection of all such G is partially ordered by inclusion. In the next few steps we will be preparing for an application of Zorn's lemma in this situation.

(6.4.6) Lemma. Let (1k, F, p) be a complete finite measure space, let G C F be a v-algebra containing all of the null sets of F, and let 6 be a density on G. Let A E F. Then there is a density 6* on the v-algebra G' generated by G and A that extends 6. Proof. Consider

C1=ICE G:,u(CnA)=0} C2=ICE G:p(C\A)=0}. Both collections are closed under countable unions, and µ is finite, so there

exist sets C1 E C1 and C2 E C2 with u(C \ C1) = 0 for all C E C1 and µ(C \ C2) = 0 for all C E C2. Now

µ(C1nC2)=,u(C1nC2nA)+u((C1nC2)\A)=0, so b(C1) n b(C2) = 0. Define A* Define 6* on g' by: 6*((B1 n A) U (B2 \ A))

(A \ b(C1)) U b(C2). Then A* - A. (6(B1) n A*) U (6(B2) \ A*)

for 131, B2 E G.

If(BInA)U(B2\A)=_ (B3nA)U(B4\A),then p((B1L B3)nA)= 0, so b(B1) A b(B3) C b(C1). Similarly b(B2) L b(B4) C_ b(C2). Since b(C1) n b(C2) = 0, we see that (6(B1),L 6(B3)) n A* = 0 and 6(B2) \ A* = b(B4) \ A*. This shows that 6* is well defined and constant on equivalence classes.

The remaining parts of the verification that 6* is a density that extends 6 are straightforward.

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285

(6.4.7) Lemma. Let (1k, be a complete probability space. Suppose .F1 C_ F2 C ... are sub-Q-algebras of .F and F1 contains all of the null sets. S u p p o s e 6, is a d e n s i t y o n fn ( n = 1, 2, ) and 6n+1 extends bn for all n. Then there is a density 6* on the or-algebra 9 generated by U° =1.Fn such that b* extends each 6n. Proof. It suffices to show that there is a lower density b on G that extends the bn. Let pn be the lifting for (Il,.fin,,u) corresponding to bn as in Proposition (6.4.3). If A E F, note that the expression pn (E-^ [IA]) is a well defined element of Goo, even though the conditional expectation [1A] is only defined up to null sets. Define, for A E 9,

b(A)={w: lim pn(E-'F"[1A])(w)=1} n-oo We will verify that b is a lower density on G that extends all bn. By the martingale convergence theorem (1.4.7), E-"'" [1A] converges a.e. to EO [1A] = 1A, so 6(A) __ A. Also, if A E Fm, then for n > m, Pn(EFn [1A])

= Pn(1A) = 16,.(A) = ]L6- (A),

so b(A) = 6,.. (A). Thus 6 extends all bn. In particular, 6([) = SZ and 6(0) = 0. It remains only to verify that b(A n B) = b(A) n 6(B). Let A, B E 9. ThenlAfB < 1A, so E-7:'" [lAnB] hi a.e. for all i, then h' > h a.e.; (3) if h is bounded, then h < p(h).

6.4. Lifting

287

Proof. We consider first the special case where µ(S2) < oo, Ihil < 1 for all i, and { hi : i E I } is directed upward. Choose a sequence hi, < hi2
h' for all i, so p(g) > p(hi) > hi everywhere. Thus p(g) > supi hi = h > g, so h is measurable, h - g, and h is the essential supremum

of{hi:iEI}. Also, p(h)=p(g)>h. For the next case, suppose µ(S2) < oo and Ihil < 1 for all i, but { hi : i E I } is not directed upward. The first case may be applied to the collection of finite suprema hi, V hi2 V ... V hi., which is directed upward.

We must check that if p(f) > f and p(g) > g, then p(f V g) > f V g. But

fVg> f, sop(fVg)> p(f) > f; similarly p(fVg)>g;;sop(fVg)> fVg. Next suppose µ(S2) = oo and Ihil < 1 for all i. Write S2 as a disjoint union Un° 1 1 of sets of finite measure, and apply the previous case on each of the sets p(Stn). Finally, in order to drop the condition 1hi I < 1, for a fixed positive integer n, apply the previous case to h A n = sup(hi A n). But this is measurable

for all n, so h itself is measurable. (Possibly h(w) = oo for some w, and possibly h ¢ Gao.)

The preceding theorem can be used to formulate uncountable versions of the standard limit theorems for the Lebesgue integral (Fatou's lemma, monotone convergence, dominated convergence, and so on) for uncountable sets { hi : i E I} of functions with p(hi) = hi.

(6.4.10) Corollary. Let { Ci : i E I } C_ F, and suppose p(Ci) Ci for all i. Then C = UiEI Ci E F and p(C) C. If { Ci : i E I} is directed upward, then p(C) = supµ(Ci). Proof. Apply Theorem (6.4.9) to the measurable functions hi = lc,.

(6.4.11) Proposition. Suppose f c G,, (52,F,µ) and h: IR, -> IR is bounded and continuous. Then p(h o f) = h o p(f ). Proof. The function f is bounded, say If I < a. Let

A={hEC([-a,a]): p(h o f)=hop(f)}. Then A contains the constants, A contains the identity function h(x) = x, and A is closed under sums and products. Also A is closed under limits of uniformly convergent sequences. By the Weierstrass approximation theorem, A = C([-a, a]).

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288

Corollary. Suppose f E Goo (Il, .F, µ) satisfies p(f) = f. If F C IR, is a closed set, then f -1(F) E Y and p(f -1(F)) C f -1(F). If (6.4.12)

G C 1R, is an open set, then f -1(G) E .F and p(f

(G))

f

(G).

Proof. Suppose G is open. The family

f={hECb(IR):0 h o f. The last application concerns "separable modifications" for stochastic processes. The result is formulated for a stochastic process with values in an interval of the line, but similar proofs can be used in other situations as well.

(6.4.13) Definition. Let (0,.F, P) be a complete probability space, and [a, b] C IR a compact interval. A stochastic process in [a, b] is a family (Xt)t of random variables with a < Xt < b. The stochastic process (Yt)tEIR is a modification of (Xt) if for all t E IR, we have Xt = Yt a.s. (the exceptional set may depend on t). A stochastic process (Xt) is separable if

there is a countable set I C IR and 1 E F such that P(Q0) = 1, and for all w E Ilo, the graph

{(t,Xt(w)) EIRx [a,b] :tEIR} is contained in the closure of

{(t,Xt(w)):tEI}. (6.4.14) Theorem. Let (Xt)tE]R be a stochastic process in [a, b]. Let p be a lifting for P). Then Yt = p(Xt) is a separable modification of (Xe) Proof. For A C R and F C [a, b], define

V(A,F)={wE ):Yt(w)EFforalltEA}. If F is closed, then by Corollary (6.4.12), we have p(Yti 1(F)) C Yt 1(F) for all t. Now for fixed F C [a, b] closed, and G C IR open, the family

{V(A,F):Afinite, ACG}

6.4. Lifting

289

is directed downward, so n V (A, F) = V (G, F) is measurable and P (V (G, F)) = inf { P (V (A, F)) : A finite, A C G }

.

So there is a countable set IG,F C G with P(V(G, F)) = P(V(IG,F, F)). Now as G runs through a countable base for the open sets in IR, and F runs through a countable base for the closed sets in [a, b], let I be the union of

the sets IG,F. So I is countable. Let no be the complement of the union of the exceptional sets V (IG,F, F) \ V (G, F).

The set no has probability 1, since there are only countably many such exceptional sets, each of probability 0. Now fix wo E no and to E R. If G is an open neighborhood of to and F is the complement of an open neighborhood of Yto (wo), we see that wo ¢ V (G, F), so wo 0 V (IG,F, F), so there exists t E In G with Yt (wo) ¢ F. As G decreases and F increases, this shows that the point (to, Yto (wo)) is in the closure of the points (t, Yt(wo)) with t E I. This shows that (Yt) is separable. Complements

The axiom of choice is used in the proofs of Proposition (6.4.4) and Theorem (6.4.8). It is known that it cannot be avoided. If p is a lifting for IR, then

f.'-' P(f)(0) is a linear functional on L,,. (IR) that is not induced by an element of Ll (IR),

so its restriction to the unit ball B of L,,. does not have the property of Baire for the weak-star topology on B (see Christensen [1974], p. 99). The map

A'-' P(lA)(0) is a finitely additive measure on a o -algebra (say the Borel sets in IR) that is not v-additive. It is not possible to prove the existence of any of these in ZF set theory without choice. R. Solovay [1970] constructed a model of ZF (with the principle of dependent choices) in which they are impossible. A lifting chooses representatives for bounded functions. It was observed

by von Neumann [1931] that a similar construction is not possible for unbounded real-valued functions. In fact, he proved more. A mapping p: L (St, -'F, p) - £ (St, .F, µ) is called a linear lifting iff: (1) P(f) = f; (2) if f g, then p(f) = p(g);

(3) if f > 0 then p(f) > 0; (4) p(f + g) = p(f) + p(g); p(af) = ap(f).

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290

(6.4.15) Proposition. Let 1 < p < oo. Suppose Gp (1k,.F, µ) admits a linear lifting. Then (12,F,µ) is purely atomic.

Proof. Let p be a linear lifting. If (1,.F,,u) is not purely atomic, then there is E E F such that 0 < µ(E) < oo and E contains no atoms. For each w E E, the map f* H p(f) (w) is a positive linear functional, hence continuous; call it uQ,. Now for each positive integer n, there exist disjoint sets Ein), E2n), , E n) C E with µ(Ekn)) = µ(E)/n. Let n

{w:p(1Ek.,))(w)=1}.

Fin>= k=1

Then F(n) = E, for all n, so

A E fn00Finl

= µ(E) > 0.

Choose W E E fl n°O_1 F(n). For each n, there exists k, 1 < k < n, such that w E so

(1 Ek,.) ,

p (1Ekn)) (w) = 1.

Then IIu.II ?

1

=

n1/nµ(E)-1/p,

1Ekn) P

which goes to oo as n --> oo. This contradicts the continuity of u,,. So in fact, (Q, .F, µ) is purely atomic. Remarks

The problem of whether a lifting exists for Lebesgue measure on the real line was proposed by A. Haar. This was solved by J. von Neumann [1931]. The existence of a lifting for a general v-finite complete measure space was proved by D. Maharam [1958]. Her proof relied on the classification of measure algebras. A more direct proof was given by A. & C. Ionescu Tulcea [1961], based on the martingale convergence theorem. The proof given here is a variant of that proof. The standard reference on liftings is the comprehensive book by A. & C. lonescu Tulcea [1969]. It contains several applications not discussed here.

7

Derivation

In this chapter we will consider the topic of derivation. We begin with the classical derivation theorems in lR and ]Rd. This can be done by considering an appropriate stochastic basis indexed by a directed set, and applying the martingale convergence theorems of Chapter 4. The theory of derivation bases in general is considered next. There are

many parallels here with the martingale limit theorems on directed sets, but we do not derive the derivation basis material from the stochastic basis material. Finally, we consider the special derivation bases known as D-bases and Busemann-Feller bases. Characterization of the bases that differentiate the indefinite integrals of all functions in a given Orlicz space Lq, occupies most of the space here.

7.1. Derivation in lR This section discusses derivation of functions defined on the space R. This is the classical theory of derivation for the Lebesgue integral. We will carry out the discussion using martingale proofs in this simple setting as an example to be followed in the later, more complicated, settings. As always, lR denotes the set of real numbers. We will write F for the valgebra of Lebesgue measurable subsets of JR, and A for Lebesgue measure on F. The Lebesgue outer measure of a set Q will be denoted A* (Q), and the inner measure by A*(Q). The outer measure may be defined as:

A*(Q)=inf{.A(A).AEF,ADQ}. Similarly, the inner measure is:

,\*(Q)=sup {\(A):AEF,ACQ}. It is known that these values may also be written as:

A*(Q)=inf{A(U):Uopen, UDQ} * (Q) = sup J .\(K) : K compact, K C Q 1.

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292

Stochastic bases

In order to apply the results from Chapter 4, we will consider some stochastic bases on R. Let a < b be real numbers. A finite subdivision of [a, b] is a finite subset of [a, b], including a and b. We write D[a, b] or D[a, b) for the set of all subdivisions of [a, b]. If t = {a = xo < x1 < < xn = b} is a subdivision, the induced partition is lrt = { [xi_1, xi) : i = 1, 2,. , n };

and the induced or-algebra is the o-algebra Ft on Il = [a, b) with atoms [xi_1, xi), i = 1, 2, , n. Note that D[a, b) is a directed set when ordered by inclusion. If tl C_ t2, we will write t1 < t2 and say that t2 refines tl. Note that t1 < t2 implies .Ftl C Ft2. Thus, l Ft)tED[a,b)

is a stochastic basis on 9 = [a, b). Suppose S is a subset of [a, b], including a and b (such as a countable dense subset). We write VS [a, b) for the subset of D[a, b) consisting of those subdivisions involving only elements of S. Then Ft) tEDs [a,b) is again a stochastic basis on SZ = [a, b). There is an easy way to generate martingales adapted to these stochastic bases. Note that we may begin with an arbitrary function F, not necessarily continuous, or even measurable.

(7.1.1) Proposition (Difference-Quotient Martingale). Let F : [a, b] -+ IR

be a function. Define simple functions ft as follows: If

t={a=xo <xl 0, so that E has one of the forms (x-r, x+r), [x-r, x+r), (x-r, x+r], [x-r, x+r]. If a > 1, then the a-halo of such an interval E is the closed interval with the same center and a times the radius: [x-ar, x+ar]. If V is a (finite or infinite) collection of nondegenerate bounded intervals, then the a-halo of V is the union of the a-halos of the intervals in V. If Q C IR and V is a collection of nondegenerate intervals, then we say that V is a Vitali cover of Q if, for each e > 0 and each x E Q, there is E E V with radius < e and x E E. Thus if x E Q fl U, where U is an open set, then there exists E E V with x E E C U.

(7.1.2) Vitali covering theorem. Let Q be a subset of IR, and suppose V is a Vitali cover of Q by nondegenerate intervals. Then (a) There exists a pairwise disjoint countable family {E,,,} C V (finite or infinite) such that

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294

(b) If \* (Q) < oo, then, for each e > 0, there exists a pairwise disjoint finite family {E1, E2, .. , Ep} C V such that

(U) En

<e.

n=1

Proof (Banach). Note that A(E \ E) = 0 for any interval E, and that if E fl F = 0 then E fl F = 0. Therefore we may assume that the intervals in V are closed intervals. We may assume also that Q # 0. First assume \*(Q) < oo. Choose an open set V Q Q with .(V) < oo. Then

Vo={EEV:ECV} is also a Vitali cover of Q. The disjoint sequence {En} of intervals will be constructed recursively. Since Q is nonempty, Vo is also nonempty. Choose El E Vo. If Q C E1, we are done; so suppose not, that is, Q \ E1 # 0. Then U1 = V \ El is open and U1 fl Q# 0. Let

61 =sup{A(E):EEV0,ECUl}. Choose E2 E Vo such that E2 C U1 and A(E2) > b1/2. Now suppose we have chosen E1i E2,

not, that is, Q \ (E1 U and Un f1 Q # 0. Let

, En. If Q C_ E1 U E2 U . . U En, we are done; suppose

U En) # 0. Then Un = V \ (E1 U . . . U En) is open

bn={A(E):EEV,ECUn}. Choose En+1 E Vo such that En+1 C Un and \(En+1) > bn/2. Note that

we have En+l flEk=0fork 0 there is 6 > 0 such that 1 IF(bi) - F(ai) I < e for every finite pairwise disjoint family {(al, bl), (a2i b2),

,

(an, bn)} of open subintervals of (a, b)

with E$ 1(bi - ai) < 6. Proof. Lebesgue measure is atomless, so uniform integrability is characterized as in (2.3.2(3)). Suppose that (ft) is uniformly integrable. Let e > 0. Then there is b > 0 such that fA I ft I dA < e for any t E D[a, b) and any set A with A(A) < 6. If {(al, b1), (a2, b2), , (an, bn)} is a pairwise disjoint family of subintervals of (a, b) with 1(bi - ai) < 6, let t consist of a, b, and all of the endpoints ai, bi. Then A = U 1(ai, bi) belongs to Ft and A(A) < 6. Therefore

i=1

F(bi) - F(ai)l = f I fti dA < e. A

Conversely, suppose that (b) is satisfied. Lets > 0 be given. Then there exists b as in (b). Let A be any measurable set with A(A) < 6. We claim that fA I ft I dA < e for all t. There is an open set U Q A with A(U) < 6. It is enough to show that fu I ft I dA < e, since fA I ft I dA < fu I ft I dA. Now U is a countable disjoint union of open intervals, so by the monotone convergence theorem, it is enough to consider the case where U is a finite disjoint union of open intervals. Choose t1 E D[a, b) containing all the endpoints of intervals in U and all the points of t.

I

F(xi) - F(xi -1)I

where the last sum is over a set of indices i such that > (xi - xi-1) _ A(U) < S, so that the result is at most e.

(7.1.11) Corollary. A function F: [a, b] -> JR is absolutely continuous if and only if it has the form F(x) = c+ f `0 f dA a

for some function f E Ll ([a, b]).

x E [a, b]

Derivation

300

Complements

(7.1.12) (Lebesgue density theorem.) Deduce from the results of this section the Lebesgue density theorem: Let A C IR be a measurable set with A(A) > 0. Then almost every point x E A is a point of density of A in the sense that for every c > 0, there is 6 > 0 so that

A(An (x-6,x+ 6)) 26

>

The Vitali covering theorem dates from 1905. The proof given here is due to S. Banach [1924]. This result is, as we have seen, closely connected to the covering condition (V) for stochastic bases. Because of this connection, the covering conditions considered in Chapter 4 are often known generically as "Vitali conditions," and condition (V) is sometimes known simply as "the Vitali condition."

7.2. Derivation in IRd The next topic is derivation in Euclidean space lRd, for d = 2, 3, . Examples of derivation bases to be considered are squares, disks, and intervals. We will write A for d-dimensional Lebesgue measure on IRd and A* for d-dimensional Lebesgue outer measure.

Substantial sets Let C be a collection of nonempty bounded open sets in IRd. Suppose C is

substantial: that is, there is a constant M such that, for every C E C, there is an open ball B with C C_ B and A(B) < MA(C). We will also assume that the sets C E C have boundary of Lebesgue measure 0: A(OC) = 0. [Recall that a point a is a boundary point of a set C C Rd if every neighborhood of a meets both C and IR.d \ C.] For example, when C is a convex set, we do have A(8C) = 0. If B is an open ball, we write r(B) for its radius.

Let V be a subcollection of C and let Q C IRd. We say that V is a Vitali cover of Q if for every x E Q and every e > 0, there is E E V with

diamE<eandxEE. (7.2.1) Theorem. Let C be a substantial collection of nonempty open bounded subsets of Ed with boundaries of measure 0. Let Q be a set in Rd, and suppose V is a Vitali cover of Q by sets of C. Then there exists a pairwise disjoint countable family {E, } C V (finite or infinite) such that A* (Q \ U-1 En) = 0. If A* (Q) < oo, then for each e > 0, there exists a pairwise disjoint finite family {E1i E2, , Ep} C V such that A* (Q \ Un=1 E") < e.

7.2. Derivation in ]R,d

301

Proof. First suppose Q is a bounded set. We apply (4.2.6) repeatedly. Let s > 0 be given. Fix an open set W D Q with A(W) < (1 + e)A*(Q). Then

W1=U{CEC:CCW} is an open set. Apply 4.2.6: there exist disjoint E1, E2,

, Ep, E C with

Ei C W and EP' 1 A(Ei) > 2-1M-13-dA(W1). Now W \ UP', Ei is an open set; let

W2=U{CEC:CCW\UP11Ei}. Repeat: there exist Ep, +1,

, E,,, E C with Ei C W \ UP11 Ei and

P2

E A(Ei) > 2-1M-13-dA(W2). i=p'+1

Continue in this way. Thus, for each k, pk

A(Ei) > [1- (1 - 2-1M-13-d)k] \*(Q)

i=1

J

Take p = A for large enough k to obtain A* (Q \ Un=1 En) < e, and allow an infinite sequence of Ei to obtain A* (Q \ U= 1 En) = 0. The case of unbounded Q may be done as before: consider Q (1 S for a disjoint family of open cubes S that cover Rd (except for a set of measure zero).

A C-partition of an open set U is a countable disjoint collection {Ci} C C, contained in U, such that

(u\uc)i =0. i=1

We write D(U) for the set of all C-partitions of U. The ordering is refinement: if

t={ C2:iEIN }

t'={ C,:jEIN }, then we say t' refines t if each Cj' is contained in some Ci. (And therefore each Ci is, except for a set of measure zero, the union of some collection of the Cil.)

We write.Ft for the v-algebra on U generated by the partition t. Theorem (7.2.1) implies that the stochastic basis (.Ft)tcv(v) satisfies the covering condition (V):

Derivation

302

(7.2.2) Proposition. Let U be a bounded open set. The collection D(U) of C-partitions of U is directed by refinement, and satisfies condition (V).

Proof. D(U) is directed by refinement: Let t1, t2 E D(U). For each C1 E ti and C2 E t2, consider the intersection w = Cl f1 C2. The collection of all C E C contained in W is a Vitali cover of W, so by (7.2.1), W is (up to null sets) the disjoint union of sets C E C. (V) is done as in (7.1.3). As usual, (V) implies convergence.

(7.2.3) Theorem. Let f be locally integrable on Rd. Then

lim J

diamEC-0 E

dA = f (x)

EE

xEE

for almost all x E Rd.

Proof. Consider the restriction of f to a large disk R. If

t={Ci:iE]N} is a C-partition of R, define

A = E (A(1) I : f dA

1ci.

iEIN

It is easy to check that (ft)tED(R) is the closed martingale E), [f I.Ft]. We may therefore conclude by (4.2.11) that (ft) converges essentially to f. Now Ll (A, R) is separable, so there is a countable collection CI that is LI-dense in

{lc:CEC,CcR}. For each positive integer n, choose a C-partition to E D(R) such that

(1) tl < t2 < ...; (2) diam E < 1/n for all E E tn; (3) A{ess sup t>tn ft > f+ 1/n} < 2-n; (4) A{ess inf t>t ft < f - 1/n} < 2-n.

Then, for each n, choose a countable family {tni}°Do such that

(1) tno = tn; (2) tni ? tn;

(3) if E E tn, and F C E, F E C1, then F E tni for some i.

7.2. Derivation in IRd

303

Now if

C. = {sup ft,,; > f + 1/n} U {inf ft,., < f - 1/n}, i

i

then A(Cn) < 2-n+1, so almost every x E R belongs to only finitely many Cn. Let Ani = UEEt., E. Then A(R \ n. 'j Ani) = 0. So almost all x E R belong to nn i Ani.

Fix a point x with x E nn i Ani and x ¢ Cn for n > no. Fix n > no. Then there is Eo E to such that x E E0; let rn be half the distance from x to the boundary of E0. Then we claim: for all E E C with x E E and diam E < rn, we have 1

A(E)

ffd,\-f(x)

n

Indeed, if E E C1, then E E tni for some i, so

(E) E f dA = tn{ (x)

f (x) + n

and similarly

A(E) E f

dA > .f (x)

-n

Now by the dominated convergence theorem, the real number 1

A(E)

LfdA

is a continuous function of E E C (as long as E C_ E0), so the inequalities hold for all E E C. This shows that sup EEC XEE

A1 4f dA- f(x)I


0, the set E T is an annulus { y E IR2 :r < I y - w I < 2r j centered at w.

is a collection B of (7.3.1) Definition. A derivation basis on deriving nets satisfying: (1) for every w E St, there is at least one deriving net (w, J, (Et)) E B;

(2) if we have (w, J, (Et)tEJ) E B and J' C_ J is cofinal in J, then we also have (w, J', (Et)tEJ') E B.

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310

If (w, J, (Et)) E B, we will say that (Et) converges to w according to B. When the derivation basis B is understood, we will write Et

w.

The sets Et that occur in a derivation basis B are known as the constituents of B. The collection of all the constituents of B is the constituency of B (or the spread of B).

Normally, throughout this chapter, a measure space (fl,

p) and a

derivation basis B will be fixed. Vitali covers and derivation

(7.3.2) Definition.

Let Q C_ Sl be a (possibly nonmeasurable) set. A collection V of constituents is a B-fine cover of Q if, for every w E Q, there exists Et = w with Et E V for all t. Let C E F be a measurable set. The collection V of constituents is a B-fine almost-cover of C if there is a set Q such that C is an outer envelope of Q and V is a B-fine cover of Q. A collection U of constituents is a full B-fine cover of Q if, for every w E Q and every deriving net Et w, there is an index to such that Et E U for all t > to. Observe that the intersection of two full B-fine covers of a set Q is again a full B-fine cover of Q; and the intersection of a full B-fine cover of Q with a B-fine cover of Q is again a B-fine cover of Q. Let y be a real-valued function defined on the constituency of B. The upper derivate of y (with respect to p) is

D*y(w) = sup limsup y(Et) t

/t(Et)

The lower derivate of y (with respect to p) is

D,,-y(w) = inf liminf Et=* w

t

-y(Et)

µ(Et)

If D*y(w) = D*'y(w), then we say -y is differentiable at w, and write D-y(w) for the common value. If -y is differentiable at almost every w E 11, then we say that y is differentiable. One family of examples of set functions y is the family of integrals. An integral is a set function y of the form 'Y(E) =

fE

f dµ,

where f is a measurable function such that the integral exists for all constituents E. Then we say that B differentiates the integral of f if Dy = f a.e.

7.3. Abstract derivation

311

Some examples of derivation bases were used in the first two sections of this chapter. In the measure space IR, for the interval basis, we postulate that En = w if En is a sequence of closed intervals, containing the point w, with positive lengths converging to 0. Then Theorem (7.1.9) can be interpreted to say that the interval basis in IR differentiates all L1 integrals. In 1R2, the centered disk basis is described by saying that En w if En is a sequence of disks, centered at w, with radius converging to 0. The disk basis is similar: En w if En is a sequence of disks, containing the point w, with radius converging to 0. Corollary (7.2.4) states that both of these bases differentiate all L1 integrals. Description of the bases corresponding to (7.2.3) and (7.2.5) is left to the reader.

For the interval basis on 1R2, we postulate that En = w iff En is a sequence of intervals, containing the point w, such that diam En -* 0. Theorem (7.2.10) shows that the interval basis fails to differentiate some L1 integrals. Many derivation bases (including all of those considered above) have a

useful approximation property. If St is a metric space, and a is a a-finite Borel measure on S2, then p is a Radon measure iff, for every Borel set B, we have

µ(B) = sup { µ(K) : K compact, K C B } , µ(B) = inf fµ(U) : U open, U 2 B } . Lebesgue measure on Rd is a Radon measure. If SZ is a complete separable metric space, then every finite Borel measure on 1 is a Radon measure (for example, Halmos [1950], (10), p. 40). Now if B is any of the bases considered above, and if C E Y, 0 < A(C)
0, then there is an open set U 2 C with A(U) < )(C) + e. However,

U={ E constituent : E C U is a full B-fine cover of C.

For abstract derivation bases we will use the following definition. A derivation basis B has small overflow if for every C E F with p(C) < 00

and every e > 0, there exist a set Co C_ C with µ(C \ Co) = 0 and a full B-fine cover U of Co such that for any A1, A2i , An E U, we have µ(U21 Ai \ C) < E. We will prove below (Proposition (7.4.2)) that many of the commonly used derivation bases have small overflow. The strong Vitali property We have seen that the Vitali covering theorem is an important tool for the classical derivation theorems considered above. Derivation bases for which corresponding properties hold will have useful derivation properties. We begin with a property for derivation bases that roughly corresponds to the covering property (V) for stochastic bases.

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312

(7.3.3) Definition. A derivation basis B satisfies the strong Vitali property if, for every C E F with 0 < µ(C) < oo, every B-fine almostcover V of C, and every E > 0, there exist finitely many pairwise disjoint constituents Al, A2, , An E V with

z<e,

(C\UA) d=1

n

µ

UAi\C)

<E.

We will prove that a derivation basis B with the strong Vitali property differentiates all L1 integrals. First, we prove a basic lemma for such integrals.

(7.3.4) Lemma. Let f E L1, and define 7(A) = fA f dµ for A E F. For any e > 0 there exists 6 > 0 so that if A1, A2i , An are disjoint constituents and C E F such that µ(C \ U Ai) < S and µ(U Ai \ C) < b, then

n

7(Ai) -7(C) <e. i=1

Proof. Since f E L1, given e > 0 there is 6 > 0 so that if G E .F with µ(G) < , An be disjoint constituents 6, then f G if I dµ < e/2. Now let Al i A2,

and C E.F with µ(C \ U Ai) < 6 and µ(U Ai \ C) < 6. Then

E 7(Ai) -7(C)I =

JA.fdP_fcfd/i fdµ_fdµdµl

=IJUA,JC

f

A ;\C

Ifl dµ+

f

C\UAi

IfI dµ

(7.3.5) Theorem. Let B be a derivation basis satisfying the strong Vitali property. Then B differentiates all L1 integrals.

Proof. Let 'y be an L1 integral; say ly(A) = fA f dµ, with f E L. We claim first that D*7 < f a.e. If not, there exist a < b such that µ* { f < a < b < D*7} > 0. Hence there is a set Q C {f < a < b < D*7} with 0 < µ*(Q) < oo. Let C be an outer envelope of Q. Now

V = ( E constituent : 'y(E) > b }

7.3. Abstract derivation

313

is a B-fine cover of Q, and therefore a B-fine almost-cover of C. Let e > 0. By the strong Vitali property and Lemma (7.3.4), there exist disjoint constituents Al, A2i , A,,, E V with u(C \ rU Ai) < e, µ(U Ai \ C) < e, I >'y(Ai) --y(C)I < e. Thus

-y(C) + E > E'y(Ai) ? b E µ(Ai) = bµ(U Ai) ? ba(C) - elbl.

This is true for all e > 0, so 'y(C) > ba(C). But f < a on C, so y(C) _ fc f dµ < ap(C) < by(C), a contradiction. Therefore D*-y < f a.e. D*ry, so we have D*ry = D*y = f Similarly, D*y > f a.e. Clearly a.e.

The weak Vitali property Next we will consider a condition analogous to condition (Vi) for stochastic bases.

(7.3.6) Definition. The derivation basis B has the weak Vitali property if, for every C E F with 0 < µ(C) < oo, every B-fine almost-cover V of C, and every c > 0, there exist finitely many constituents A1, A2, with

,

A,,, E V

(a) Fi (C \ U 1 Ai) < e,

(b) En µ(U 1Ai\C)<E,n

1z(Ai) - i (U 1 Ai) < e. (The left-hand side of (a) is called the "deficit"; the left-hand side of (b) is called the "overflow"; the left-hand side of (c) is called the "overlap.") There are some useful alternative formulations of the definition. (c)

1

(7.3.7) Proposition. Let B be a derivation basis. The following are equivalent

(1) The weak Vitali property: For every C E F with 0 < µ(C) < 00, every B-fine almost-cover V of C, and every e > 0, there exist finitely many constituents A1, A2i , A,, E V with a (C \ U Ai) < E, /2(UA2\C) <E, E F with 0 < u(C) < oo, every B-fine almost-cover V of C, and every e > 0, there exist finitely many constituents Al,A2,...,A,,EVwith

Il>1Ai-1CII1<e.

(3) For every C E F with 0 < µ(C) < oo, every B-fine almost-cover V of C, and every e > 0, there exist countably many constituents A1, A2, ... E V with p (C \ U Ai) = 0, µ (U Ai \ C) < e, E u(Ai) p (U Ai) < e.

(4) For every C E F with 0 < p(C) < oo, every B-fine almost-cover V of C, and every e > 0, there exist countably many constituents E V with 11 1Ai -1cII1 < E. Al, A2,

Derivation

314

Proof. (1) (3). Let C, V, e be given. Write e' = e/3. Apply (a) to obtain finitely many constituents All, A21, E V with µ (C \ U Ail)
E'y(Ai) >_ b E p(Ai) > bp(C) - eIbl.

This is true for all E > 0, so 'y(C) > by (C). But f < a on C, so y(C) _ fc f dp < ap(C) < bp(C), a contradiction. Therefore D*y < f. Similarly, D*y > f. (a) (d). Suppose the density property holds. Let V be a B-fine cover

of Q and C an outer envelope of Q; suppose 0 < µ(C) < oo; let e > 0. Choose a with 0 < a < 1 such that

0 < ( 1a \\

- 1) µ(C) < E.

Now if Y C Q and µ* (Y) > 0, let

V(Y,a)={EEV:,u*(YnE)>ap(E)}, ry=sup {µ(E):EEV(Y,a)}. From the density property applied to an outer envelope V of Y, we see that some point of Y is a point of density of Y, so there exists w E Y and Et

w, with Et E V, so µ(Y fl Et)/p(Et) -+ 1 > a. Thus V(Y,a) # 0,

and therefore ry > 0. If Y C Q and t* (Y) = 0, write ry = 0. Now fix ,3 with 0 < 0 < 1. Let X1 = Q. Then p* (Xl) > 0, so rX, > 0.

There exists Al E V with µ(A1) > 3rx, and µ(C fl A1) > a t(Ai). Let

7.3. Abstract derivation

317

X2 = X1 \ A1. Continue recursively: Suppose A1, A2, .. , An E V have been defined such that

µ(Xi n Ai) > ali(Ai),

u(Ai) > ,3rx.,

where Xi+1 = Xi \ U;=1 Aj and Xi is an outer envelope of Xi. If we have u*(X,,+1) = 0, then the recursive construction stops. Otherwise, let An+l E V satisfy µ(A,,+1) > and u(X,,+1 n An+1) > aµ(An+1) So we get a (finite or infinite) sequence of sets A1, A2i E V such that Ai n Xi are disjoint subsets of Q, p(C) > µ(C n U Ai) ? /L( U(Ai n Xi))

_ E lL(Xi n Ai) > a E ti(Ai) Thus E p(Ai) < (1/a)p(C n U Ai) < oo. Now we claim that µ(C \ U Ai) = 0. If the sequence Ai is finite, then p*(XN+1) = 0 for some N, so µ* (Q \ UN 1 Ai) = 0 as claimed. So suppose

the sequence Ai is infinite. Then /3>irx;
0 so that e' < e and µ(D) < 2E' implies II1DIIb < E/2. Then by (2.1.22) choose 6 < E/2 so that if g has integer values and IIgIIp < b, then IIgII1 < C. Now suppose C and V are given. Then by (1) there exist A1, , A,,, E V 1A.-1CII,D < 6. Now lUAi\c < I 1Ai-1C I, SO II1UA;\CII'J' < 6, with II > and thus µ(U Ai \ C) < e' < E. Similarly, 1c\UA; < I > 1A: - 1C 1, so µ(C \ U Ai) < E' < E. Finally, µ(C A U Ai) < 2e', so II1cjUA, III < E/2, and thus I < I E 1Ai - 1CI + II1C - 1UA.II,p 0 so that 8
lA. - 1UA, IIi, < 6. Therefore

II1C\UA; II4 and %F be conjugate Orlicz functions, 4> finite. Let f E Lq,, and define -y(A) = fA f dµ for A E F, µ(A) < oo. Let A1, A2i

,

A be constituents and let C E .F. If II E 1Ai - icIlz < E, then

7(Ai) -7(C) < 2eflfII,. i=1

Proof.

I'Y(Ai) - y(C) I = If (> 1A; -1C)

0, still %F(cp(0)) = 0.) Thus Dl # 0, where

Dl = { B E V : f cp(0) dp + p(1)p(B \ C) < /3µ(B) } .

l

B

J

Choose B1 E D1i µ(B1) > (1/2) sup { µ(B) : B E Dl }. Then we write Al = {B1} so that Al satisfies (i), (ii), and (iv). If Al does not satisfy (iii), the recursive construction stops. Otherwise continue. Suppose Ak = {B1, B2i , Bk} is defined, satisfying (i)-(iv). Then Dk+1 # 0, where Dk+1 = j B E V :

L co(nAk) du + o(1), (B \ C):5 ,3 (B) } .

Choose Bk+l E Dk+1 with p(Bk+1) >- (1/2) sup { µ(B) : B E Dk+1 }. Then let Ak+1 = Ak U {Bk+1}. Thus Ak+1 satisfies (i), (ii), and (iv). If it does not satisfy (iii), stop. Otherwise continue. We therefore get A = Uk Ak = {B1, B2i }, a finite or countably infinite collection of constituents. We claim that A has the properties required

Derivation

322

by (V,), namely: µ(C \ QUA) = 0, p(UA \ C) < e, IleAII4' < C. Each Ak satisfies (i), so (by monotone convergence) so does A:

1 P(est) dp < 71µ(C n UA) < 71µ(C) < el. By the choice of el, we have IleAII4, < e. Each Ak satisfies (ii), so A does also. Therefore

(1 - 9) µ(UA) < (1 - 77) J nA dp < µ(C n UA) < oo. Also,

p(UA \ C) < p(UA) - µ(C n UA) < (1 - (1 - 71)) p(UA) = 71,u(UA)


0. Thus p(B) < 2p(Bk) for all k. But > p(Bk) < oo, a contradiction, so µ(C \ QUA) = 0. Therefore (V4,) is verified.

Property (FV4,)

Next we come to a Vitali type of property, with the advantage that condition (A2) is not required for the converse. Let 4) be an Orlicz function. A derivation basis has property (FV4,) if, for every e > 0, every C E F with 0 < p(C) < oo, and every B-fine almost, An E V and nonnegative cover V of C, there exist constituents Al, A2i scalars al, a2i , an such that

ai1A,-1C i=t

<e, 4,

aip(Ai) - p(C) <e. It is easy to see that (V4,) implies (FV4,) with ai = 1.

7.3. Abstract derivation

323

(7.3.18) Proposition. Suppose L4. C_ L1. Then B has property (FV4.)

if and only if for every e > 0, every C E F with 0 < µ.(C) < oo, and , An E V, and every B-fine almost-cover V of C, there exist A1, A2i a1, a2,

, an > 0 such that 11E ai 1A: - 1C II4, < E.

Proof. If B has property (FVp), then clearly the other condition holds. For the converse, suppose the condition stated is satisfied. For given e > 0,

there is 6 > 0 so that 11f II. < 6 implies 11f II 1 < e. We may also as, A, E V, and sume 6 < e. Then given C and V, there exist A1, A2i a1 i a2, , an > 0 such that II > ai 1Ai - 1c 114, < 6 < e. Then we know that II E ai 1Ai -1C111 <E, so

IE ai,u(Ai)

- µ(C) I= IfJ (> ai 1Ai - 1C) dµ I


0 with II

ai 1A, -1cliw _ b E aiµ(Ai) > bp(C) - Ible. This is true for all e > 0, so y(C) > bp(C). But f < a on C, so

y(C) =

J f dµ

ap(C) < b(C)

a contradiction. Therefore D*y < f . Similarly, D*y > f.

Derivation

324

For the converse, we first take the case L4, C L1.

(7.3.21) Theorem. Let 4) and 41 be conjugate Orlicz functions with 4 finite and L4. C L1. Suppose B differentiates all LT integrals. Then B has property (FV4,).

Proof. Suppose (FV4,) fails. There is e > 0, Q C_ 1, and a B-fine cover V of Q such that, if C is an outer envelope of Q, ai > 0 and Ai E V satisfy

I Eaiµ(Ai) - µ(C)I < E, then

11E ai 1A: - lc

(7.3.21a)

I

> E.

We assume also that E < µ(C). By (2.1.23), there is e' > 0 so that If I < 1, 11f 11, < e' imply If 114, < e/2. Assume also that c' < E. Since L4, C L1, there is s" > 0 so that IIf114 < E" implies I1f111 < e'/2. Assume also that E" < E/2. We begin with an application of the Hahn-Banach theorem (5.1.2). Consider three subsets of L.,:

Cl = {a i 1A:: ai > 0, Ai E V, I E.aiµ(Ai) - µ(C)I < E'/2 1,

C2=EL4,:. 1}. Now

-2
1/(µ(C) - e'/4) for all w E Q. Now if D*-y = f a.e., we would have f (w) > 1/(µ(C) - E/4) a.e. on C. But 1c E C2, so we would have 1 > 'y(C)

=ff

dµ

1,

µ(C)(CE'/4 > a contradiction. Thus D*y = f fails on a set of positive measure. Therefore B does not differentiate the integral of the Lq, function f.

(7.3.22) Next we should consider the case L4, V= L1. Let 4) and T be conjugate Orlicz functions. Suppose B differentiates all L,, integrals. If A E F, u(A) < oo, then 1A E Lq,. So B has the density property, and therefore B differentiates all L,,. integrals. Thus B differentiates integrals of functions in L,v + L. We have seen in Proposition (2.2.13) that Lp + L,,. is itself an Orlicz space L*a, where AI's is the shifted Orlicz function given by (7.3.22a)

ty(v -

0,

v 1.

A calculation shows that the conjugate Orlicz function is then (7.3.22b)

4)g(u) = 4)(u) + u,

and LD. = L4, fl L1. Since L4,e C L1, the case of (% and '$ is covered by the previous material. (4) is finite if and only if the derivative cp is finite, if and only if Eli is unbounded, if and only if lim'1(v)/v = oo.) Thus we have proved:

Theorem. Let ' be a finite Orlicz function with lim'(v)/v = oo. Let IQ8 and 4)$ be defined by (7.3.22a) and (7.3.22b). Then the following are equivalent: (1) B differentiates all L,, integrals; (2) B differentiates all L p. integrals; (3) B has property (FV4, ).

The important case not yet completed is Lp = L1, Lt = L. We will discuss this in the next section.

326

Derivation Complements

(7.3.23) (Stochastic basis.) Let (fl,.F, P) be a probability space, let J be a countable directed set, and let (Ft)tEJ be a stochastic basis consisting of finite a-algebras. For each t E J and w E S2, let Et (w) be the atom of Ft containing w. We may define a derivation basis B on SI by postulating (Et(w))tEJ,

w

for J' C J cofinal. Now if X E L1, then there correspond a martingale

Xt=E''[X] and an integral

y(A) = fXdP. Now of course

'y(Et(w))

= Xt(w), P(Et(w)) so Dry = X a.e. if and only if Xt --> X a.s. The analogies between the stochastic basis covering properties of Chapter 4 and the derivation basis Vitali properties of this chapter can be exhibited in this setting. For example, (.Ft) satisfies the covering condition (V) if and only if B has the strong Vitali property. (7.3.24) (A "demiconvergence" result.) Suppose a derivation basis B has the weak Vitali property. Then B "lower differentiates" all nonnegative L1 integrals. That is: if f > 0, f E L1i and y(A) = fA f dµ for all A E F, then D.y > f a.e. (If f > 0, we may dispense with Lemma (7.3.8) in the proof of the first part of (d) (c) in Theorem (7.3.11). See Hayes & Pauc [1970], Proposition 2.1, page 19.) (7.3.25) (Overflow.) Talagrand's definition for property (V4.) (Talagrand[1986]) is as in Proposition (7.3.12) part (2), except that the overflow

condition µ(U Ai \ C) < e is omitted. Of course, if B has small overflow, then the condition is not needed. But for completely general B it is needed. For example, suppose we have the trivial derivation basis defined on [0, 1] so that En = w if En = [0, 1] for all n. It is easy to arrange zero deficit and overlap, but the integral of no nonconstant f E L4. has derivative f. (7.3.26) ((FV4.).) Talagrand's definition for (FV4.) (Talagrand [1986]) is: for every e > 0, every C E F with 0 < u(C) < oo, and every B-fine almostcover V of C, there exist constituents A1, A2, , An E V and nonnegative scalars a1, a2i , an such that

ai1As -1AEai1A,II <E, aiµ(Ai) - u(C) I < E.

7.4. D-bases

327

For general derivation bases, this is in fact not sufficient for differentiation of L41 integrals. For example: Let (1, F, p) be [0, 1] with Lebesgue measure.

Enumerate the rationals in (0, 1) as rn. Let En = [0, rn], n E IN, and postulate En w for all w E [0, 1]. Then certainly B differentiates the integral of no nonconstant f E L*. But if V is a B-fine almost-cover of any set C, and E > 0, then there exist arbitrarily large n with I rn - A(01 < e,

soEnEV. Now let Al=En and al=1, soIla,1A,-1Aa,1A,IIt=0 and Iaip(Al) - p(C)I < e. (7.3.27) (Both inequalities are needed in (FVD).) Let _

ai 1Aj. Then III - lc 112 small does not imply I E ai p(Ai) - p(C) I is small. Let (fI, F, p) be IR with Lebesgue measure. Take C = [0, 1], Ai = [0, 1] U [i, i + 1] and ai = 1/n f o r i = 1, 2, n. Then II ai 1 A; - lc 112 = 1/V/n_ but I

.aip(Ai) - p(C)I = 1.

Remarks

A reference on derivation theory is Hayes & Pauc [1970]. Much of the material in this section follows their treatment. Theorem (7.3.17) on the necessity of (V..) is due to C. A. Hayes [1976]. We have used his proof. Property (FV4,) is due to Talagrand [1986] (note (7.3.26)).

7.4. D-bases The most often used derivation bases are of special kinds. In this section we will consider the D-bases and a still more special case, the BusemannFeller bases. D-bases

Let (S2, .F, p) be a or-finite measure space. A D-basis on (S2, F, p) is a pair (£, 6), where 6 is a family of measurable sets E with 0 < p(E) < oo, and 6 is a function 6: 6 -+ (0, oo), such that, for every w E St and every E > 0, there exists E E £ with w E E and 6(E) < E. Given a D-basis (£, 6), we may define a derivation basis by specifying En = w iff w E En E £ and 6(En) -> 0. A derivation basis B that can be specified in this way will also be called a D-basis. We may even write B = (£, 6). For example, Il may be a metric space, with metric p, and 6(E) may be the diameter of E:

6(E) = diam E = sup { p(x, y) : x, y E E 1.

Some of the derivation bases in Sections 7.1 and 7.2 are D-bases with 6 as diameter: the interval basis in IRd; the disk basis in IR2 (but not the centered disk basis). We have seen a correspondence between stochastic bases and derivation bases.

The condition of a countable cofinal subset for stochastic bases

corresponds roughly to being a D-basis for derivation bases.

Derivation

328

The measurability of the upper and lower derivates is true for certain D-bases.

(7.4.1) Proposition. Let f be a metric space, let p be a Radon measure on p, let (E, 6) be a D-basis where E is a collection of open sets, and 6 is diameter. If y: E --+ lit is any set function, then D*y and D*y are measurable functions.

Proof. Fix a real number t. For positive integers m and k, let

Pmk=US EE£:6(E)
t+m } p(E)

00

{D*y > t} = U n Pmk m=1 k=1

is a measurable set (a Gba set). This is true for all t, so D*y is a measurable function (a function of the second Baire class). Similarly D*y is measurable. The next result shows that the most common derivation bases have small overflow.

(7.4.2) Proposition. Let St be a metric space, let u be a Radon measure, and let (E, 6) be a D-basis, where 6 is diameter. Then (E, 6) has small overflow.

Proof. Let C E F, 0 < µ(C) < oo. Let e > 0 be given. There is an open set U D C with µ(U) < p(C) + e. The family

U={EE£:ECU} is a full B-fine cover of C, and if Al, A2, µ(U \ C) < C.

, An Elf, then p(U Ai \ C)

Properties (A) and (C)

The derivation basis B has property (A) if for every C E F with 0 < µ(C) < oo there is a constant M such that for every B-fine almost-cover V of C and every e > 0, there exist Al, A2i , An E V with u(U Ai \ C) < e, and E 1A; 11

< M E p(Ai)

7.4. D-bases

329

The differentiation theorem follows the usual outline.

(7.4.3) Theorem. Suppose B has property (A). Then B differentiates all L1 integrals.

Proof. Let f E L1, and define y(A) = fA f dµ. We claim f > D*y. If not, then,u*{ If < D*y} > 0, so there exist a < b with µ*{ If < a < b < D*y} > 0.

LetQC{f D*y. Similarly f < D,,-y.

For the converse, we use the Hahn-Banach theorem, but only on a finitedimensional space.

(7.4.4) Theorem. Suppose the D-basis B = (E, b) differentiates all L1 integrals. Then B has property (A). Proof. First, B differentiates all L1 integrals, so B has the density property, and therefore the weak Vitali property. In particular, B has small overflow.

Let C E F with 0 < µ(C) < oo. Suppose property (A) fails. Choose Mk T oo with 00

1

Mk

< µ(C) 2

Then for each k, there exists a B-fine almost-cover V of C such that A1, A 2 ,--- , A,,, E Vk implies I) E 1A; IIa, > Mk E µ(Ai). We may assume

6(E) 0, then IIEai1A;11. > Mk Eaip(Ai). This may be seen by approximating the scalars ai with rational numbers, multiplying through by a common denominator, and observing that repetitions are allowed in the list Al, A2, An. By the weak Vitali property, there exist finitely many sets Alk, A2k,

E Vk with

p(Ui Aik \ C) < p(C)/2 and u(C \ Ui Aik) < p(C)/2. Let Ak be the finite algebra on Q generated by the sets Alk, A2k, . Consider two subsets of L. (ul, Ak, P):

Cl = > ai IA,,, : ai

aip(Aik) = 1

0,

,

C2={eEL.(Ak):1; <Mk}. These sets are disjoint, convex, and closed. C1 is compact. Thus there is fk E L1(ci, Ak, p) with f t; fk dµ > 1 for all l; E Cl and f t; fk dµ < 1 for all l E C2. Because of C2, we have fk > 0 and IIfkDII p(Aik) We may therefore construct such a function fk for each k. Let f = >k fk. Then f > 0, and IIf 111 E 1/Mk < oo, so f E Ll. But we claim B does

not differentiate the integral y of f. Let

B=1

IUAik

k

i

Then p(C \ B) < >k p(C \ Ui Aik) < p(C)/2, so p(C fl B) > p(C)/2. For every w E B and every k, there is E E Vk C_ £ with w E E, 6(E) < 1/k and

fE f dp > fE fk dp > p(E). Therefore D*ry > 1 on B. But if f > 1 on B, then p(2) < E Mk < µ(2 ) ' < p(B) ` JB f dp f on a set of positive measure. There is another property closely related to property (A). The derivation basis B has property (C) if, for every e > 0, there exists a constant M such that if C E F, e < p(C) < oo, and V is a B-fine almost-cover of C, then for every 77 > 0 there exist A1i A2, , An E V with II E 1A; IIw < Mp(U Ai)

andp(UAi\C) 0, there exists a set Co C C with p(C \ Co) = 0 and a full B-fine cover U of Co such that, for any Al, A2, , An Elf, we have p(U Ai \ C) < 77.

7.4. D-bases

331

Now if B has small overflow, property (C) may be stated: For every > 0, there exists a constant M such that if C E F, E < u(C) < oo, and , A,,, E V with V is a B-fine almost-cover of C, then there exists A1, II E 1Ai 11. < Mu(U Ai). Indeed, if this is true and q > 0 is given, choose Co and U as in the definition of small overflow; then E < u(Co) < oo and V flU is a B-fine almost-cover of CO, so that there exist A1, , A E V flU with II E 1Ai Iloo < Mu(U Ai) and µ(U Ai \ C) < 17. Property (A) may be similarly simplified. Properties (A) and (C) are equivalent for a D-basis with small overflow:

(7.4.5) Theorem. Let B = (E, b) be a D-basis with small overflow. The following are equivalent:

(1) Property (A): for every C E F, µ(C) > 0, there is M such that for every B-fine almost-cover V of C, there exist A1, A2, with 11 > 1A; II. < M 1: µ(Ai).

,

A,,, E V

(2) For every C E F, µ(C) > 0, there is M such that for every Bfine almost-cover V of C, there exist A1, A2i , A,, E V with II E 1A, II. < Mµ(U Ai)

(3) Property (C): for every E > 0 there is M such that if C E F, µ(C) > E, and V is a B-fine almost-cover of C, then there exist A1, A2, ... , A E V with II E 1Ai 11m < My(U Ai)

Proof. (2) (1)

(1) and (3) = (2) are easy. (2). Since (A) holds, B differentiates all L1 integrals, so the

density property holds: that is, the weak Vitali property holds.

But suppose (2) fails. There is C E F, µ(C) > 0, such that there ex, A,, E Vk implies ist B-fine almost-covers Vk of C for which A1i A2, II E 1Ai Iloo > 2kpi(U Ai). Now for each k, use the weak Vitali property to choose finite sets 13k = {B1ki B2k, } _C Vk with b(Bik) < 1/k,

Ei µ(Bik) -,u(Ui Bik) < 2-k, µ(Ui Bik \ C) < 2-k, and µ(C \ Ui Bik) < 2-k. Note that E µ(Aj) - a(U Aj) < 2-k also holds for any subfamily {Aj } of Bk. Now C = lim supk Ui Bik satisfies p(C \ C) = 0, and Uk 13k is a B-fine cover of C. _

We may apply (1) to the set C, to obtain a corresponding constant M. Choose j so large that 2j-2 > M. Now V = U00 13k is a B-fine cover of , An E V with II > 1Ai 11. < M E p(Ai). Each C, so there exist Al i A2, set Ai belongs to some 13k with k > j; call it 13k(i). Now n

L i=1

00

ju(Ai)

=k=jEk(i)=k E N'(Aj)

Derivation

332

k=j

00

00

E2 -k+1

k=j

i=1

< 2-j+2

00

E 1A= i=1

00

a contradiction. Hence (2) holds. (2) (3). Suppose (2) holds. Then (A) holds, so again the weak Vitali property holds.

Suppose (3) fails. Then there exist e > 0, C,,, E F with µ(Cm) > e, and B-fine almost-covers Vm of Cn, such that A1, A2i , An E V. implies 1A; 1100 > 2mµ(IJ Ai). For each m, choose by the weak Vitali property a finite set 13.,,, = {Bim, B2m, } C V. with S(Bi,m) < 1/m, 11

/L(Cm \ Ui Bim) < 2-m, and µ(Ui Bim \ Cm) < 2-m. Now let C = lira supra IJi Bim. Then Um=113m is a B-fine cover of C and p(C) > e. Apply (2) to C to obtain a constant M. Choose j so that 2j-1 > M. Then V = Um=;13m is a B-fine cover of C. Thus there exist A1, A2i , An E V 1Aj 11ao < Ma(U Ai). Each set Ai is in some Bm(i). Now with 11

µ

(UA)n i=1

µ

m=j

U

Ai

m(a)=m

00

E 2-m E 'A, m(i)=m j=m

00

< 2-j+1 i=1

00

a contradiction. Hence (3) holds. Halo theorems

The classical derivation theorems, such as the theorems in Sections 7.1 and 7.2, use the Vitali covering theorem. Banach's proof of this theorem uses a simple geometric fact about "halos" of sets in W. It is useful to generalize this idea. We will use the "essential supremum" or "essential union" of a family

of sets (see Section 4.1). In many cases, this is the same as the ordinary union:

7.4. D-bases

333

(7.4.6) Proposition. Let St be a metric space and let µ be a Radon measure on Q. Let (Ai)iE7 be a family of open subsets of Q. Then

A=UAi iEI

is an essential supremum of the family.

Proof. Certainly A is open (hence measurable) and A D Ai for all i. Suppose also B is measurable and B D Ai a.e. for all i. We claim that B D A a.e. Suppose not. Then µ(A \ B) > 0. Thus there is a compact set K C_ A\B, with µ(K) > 0. Now (Ai)iEI is an open cover of K, so there is a finite subcover, Ai, UAi2 U . UAi

K. Then K C (Ai, UAj2 U. .

\B.

In fact p(Ai \ B) > 0 for some i, which contradicts B D Ai a.e.

Let B = (£, 6) be a D-basis, let A E £, and let a > 0. The a-halo of A is

H(a, A) = ess sup {EE£:EnA#0, 6(E) 0, choose no so that p(Cno) < e/2; then choose ko so that µ(S(a, %o, Cno)) < e. If n > max{no, ko}, then µ(S(a, zn, Cn)) < E. This shows that limn µ(S(a,1)n, Cn)) = 0. Thus B has the weak halo evanescence property. Conversely, suppose that B has the weak halo evanescence property. Recall (Proposition (7.4.1)) that the upper and lower derivates are measurable

functions under the conditions assumed. Let C E F, 0 < µ(C) < 00. We want to show that almost every point of C is a point of density of C. Let -y be the integral of 1C: we want to show that D*y > 1 a.e. on C. Suppose not. There exists a > 0 such that µ { w E C : D,,-y(w) < 1 - a } > 0. Thus there is a compact set K c c n {D,,y < 1 - a} with µ(K) > 0. Now V1

={ E

µ(KnE)

i A(9i(Sm)) < 2, and 6(9i(Sm)) < 1/m. Let

Cm = Ui 9i(Bm). Then: for almost every w E J, there exists a con-

stituent E with w E E, 6(E) < 1/m, and A(E fl Cm) > ap(E). Also, A(Cm) < >i A(9i(Bm)) < 4-m E {G(9i(sm)) < 2.4--. Now let C = U,°_1 Cm. Thus A(C) < > A(Cm) < > 2.4-m < 2/3. We claim that B does not differentiate the integral ry of 1c. For almost every w E J and every m, there is a constituent E with w E E, 6(E) < 1/m, and A(E fl C) > A(E fl Cm) > aA(E). Therefore D*ry(w) > a, so D*. = lc fails on the set J \ C, which has positive measure.

7.4. D-bases

339

For derivation of L1 integrals, we should consider a refinement of (WH). We postulate not only for each a there is a constant M, but more precisely

how M depends on a as a - 0. The derivation basis B has property (WH1) if there is a constant K such that, for every a, 0 < a < 1, and every C E F, we have A(S(a, C))
Proceeding as in (7.4.11), we obtain sets Cm contained in the unit interval J of 1R,d such that )(C,,,.) < 1/(Km'P(1/a,,,.)), and for almost every w E J, there is a constituent E with w E E, 6(E) < 1/m and a,,.A(E). Now consider the nonnegative function

f =sup-1Cm, m am

and its integral -y(E) = fE f dA. We have dA

(7.4.12b)

=E'\(C,»)`p( 1)

a0A(E). Thus 'Y(E)

fE f dA > JE m 1Cm dA = m A(Cm fl E) > A(E).

Derivation

340

Thus D*y(w) > 1 on J. But if f > 1 on J, then we would have f T (f) dA > f f'Y(1) dA = T(1), which contradicts (7.4.12a) and (7.4.12b). Therefore

A(f 0. For a converse result, we will consider a "functional" variant of condition

(WHq,). The derivation basis B has property (FHp) if there is a constant K such that, for all disjoint bounded sets C1i C2, , C, E Y and all , c,,, we have

nonnegative scalars c1, c2,

µ (ess sup { E constituent : E cip(Ci n E) > µ(E) })

< K > IF(ci.)p(Ci) Note that (FH p) implies (WH p) by taking a single set C1 and scalar cl = 1/a. On the other hand, if (FH,,) holds, then the inequality remains true even for infinite lists C1, C2, and c1, c2, . Condition (FH4,) is more complex than (WH*), but it is necessary and sufficient for differentiation of L,1, integrals.

Note an alternative formulation: If we write g = ci lei (a nonnegative simple function), then the inequality > ci.µ(Ci n E) > p(E) becomes fE g d1i > µ(E), and > IF (ci) p(Ci) becomes f W (g) dµ.

(7.4.13) Theorem. Let B be a Busemann-Feller basis, and let 4' be an Orlicz function satisfying (02) with 0 < '(v) < oo for all v > 0. Then B differentiates all Lw integrals if and only if B has property (FHq,).

Proof. Assume (FH,,). Property (FHq,) implies (WHq,), which implies (WH). Thus (7.4.11) B differentiates the integrals of all Lm functions. Let f > 0 belong to Lw, and let y(E) = fE f dA for constituents E. We claim that Dry = f a.e. Write Ci = {i - 1 < f < i}; the sets Ci are disjoint. For m E IN, let f,,,, = f 1 [f 0. Now f E Lip, where IF satisfies (02), so f oP(2 f /e) dA < oo.

On the set Ci, we have (for i > 2) i/e < 2(i - 1)/e < 2f /e, so 00

i=2

4,

(cJL) i A(Ci n E) > e,\(E) i=m+1

The sets decrease: Sl Q S2

.

By condition (FHp),

\(Ci);

A(S,,,) f (w) and w ¢ Sm. Now

-

°° m y(En n Ci) y(EE) \(En) A(En) = + i=

y(E,,. n Ci) A(En) 1

By the definition of S,,,,, 00

00

y(EnnCi)

i=

iA(E,,,nCi)


f (w), so f (w) = fm(w) and M y(EE n Ci) = ym(En) A(En) A(EE) 1

L. i=1

fm (w)

Thus

f (w) - e < liminf (En) < lim sup y(En) < f (W) + e. This is true for all En = w, so .f (w) - e < D.-y(w) < D*y(w) < f (w) + e.

Finally, since a was arbitrary, we have f (w) = Dy(w) a.e. For the converse, suppose (FHq,) fails. Choose Km T oo with °O

2

EK

(7.4.13a)

M=1

< lY(1).

m

There exist nonnegative simple functions gm, with bounded support, such

that A (U { E : fE gm dA > A(E) })

>

Km f W (gm) dA.

For each m, there is a finite union Sm of constituents E with fE gm d,\ > A(E) such that )1(Sm) > Km f %P(gm) dA. Let J be the unit interval in Rd. There exist homotheties 01i 02, such that Ui Oi(Sm) C J, S(6i(Sm)) < 1/m, A(J \ Ui Oi(Sm)) = 0, and >i A(Bi(Sm)) < 2. Thus if fm = supi gm o ei, we have

f T (.fm) dA
,,,, v log v. In particular, B does not differentiate all L1 integrals. This is essentially the fact that is used in the counterexample (7.2.10).

M be

Figure (7.4.14).

A(S(a, J)) = 1 + (4/a) log(1/a).

Figure (7.4.15a). Weak halo.

7.4. D-bases

Figure (7.4.15b). Subdivide.

343

Figure (7.4.15c). Translate.

(7.4.15) (Rectangle basis.) In 1R2, the rectangle basis is the BusemannFeller basis consisting of the (open) rectangles, with sides not necessarily parallel to the axes. This basis fails the density property. This can be seen using Theorem (7.4.11) as follows. The weak halo S(1/2, C) of a triangle C with vertices a, b, c (shown in Figure (7.4.15a)) contains a triangle with vertices a, b', c', where b is the midpoint of the line segment from a to b' and c is the midpoint of the line segment from a to c'. Consider a triangle (shaded in Figure (7.4.15b)) and the corresponding larger triangle (white). Subdivide it into many smaller triangles as shown. Then translate the triangles as in Figure (7.4.15c). The weak halo S(1/2, C) of the shaded

portion C contains the white portion. By taking a large number of subdivisions and using appropriate translations, it may be arranged that the shaded area is as close to 0 as we like, while the white area remains large. This shows that the property (WH) fails. (Details of this construction are in Busemann & Feller [1934]. Or see Hayes & Pauc [1970], Section V.5, p. 104.) Remarks

D-bases, Busemann-Feller bases, and many other variants are discussed by Hayes and Pauc [1970]. The reader may consult that volume for further information. In the term "D-basis," the "D" is for Denjoy. See Denjoy [1951], Haupt [1953], Pauc [1953]. The general theory allows the function 6 to be something other than the diameter. There is also the possibility of using another "disentanglement function" in the definition of the halo. Property (A) is the derivation version of a condition of Astbury [1981b]. The equivalence of property (A) and property (C) (for special D-bases) is proved in Millet & Sucheston [1980c]. They also prove the sufficiency of property (C) for differentiation of Ll integrals. Necessity of property (C) for differentiation of Ll integrals is stated in Talagrand [1986]. Busemann-Feller bases come from a paper of Busemann and Feller [1934].

Pointwise ergodic theorems

In this chapter, we will prove some of the pointwise convergence theo-

rems from ergodic theory. The main result of this chapter is the superadditive ratio ergodic theorem. It implies the Chacon-Ornstein theorem, the Kingman subadditive ergodic theorem, and, for positive operators, the Dunford-Schwartz theorem and Chacon's ergodic theorem involving "admissible sequences." We consider positive linear contractions T of L1. Our plan is as follows. We first prove weak maximal inequalities, from which we obtain the Hopf decomposition of the space SZ into the conservative part C and the dissipative part D (8.3.1). Assuming T conservative (that is, Q = C), we prove the Chacon-Ornstein theorem (8.5.4), i.e., the convergence to a finite limit of the ratio of sums of iterates of T applied to functions f and g. The limit is identified in terms of f, g, and the o -algebra C of absorbing sets. The superadditive operator ergodic theorem is proved in the conservative case (8.4.6). It is then observed that the total contribution of SZ to C is a superadditive process with respect to the conservative operator Tc induced by T on C . Since the behavior of the ergodic ratio on the dissipative part D is obvious, the Chacon-Ornstein theorem (8.6.10) and, more generally, the superadditive ratio theorem (8.6.7) will follow on 0. This affords considerable economy of argument, since the direct study of the contribution of D to C is not obvious even for additive processes. The superadditive theory (or, equivalently, the subadditive theory) is mostly known for its applications, but in fact the notion of a superadditive process is shown to shed light on the earlier additive theory of L1 operators. Throughout this chapter, the term "operator" means a bounded linear transformation. An operator T defined on (equivalence classes of) realvalued measurable functions on Il is called positive if f > 0 a.e. implies T f > 0 a.e. The operator T is a contraction on LP if JjT f lip < 11f ill, for each f E L7, . A positive operator T is called sub-Markovian if it is a contraction on L1. A positive operator T is called Markovian if it preserves

the integral, f T f dp = f f dp for f E Li , so that T is an isometry on Li . If f and g are extended real-valued functions, we write f V g for the pointwise maximum, f A g for the pointwise minimum, f + = f V 0, and

f-=(-f)V 0.

Throughout Chapter 8, we will let (S2, F, p) be a o-finite measure space, and T a positive contraction on L1(SZ, F, µ). Observe that a positive linear operator on L1 can be extended to act on the lattice of all extended real-valued functions f such that the negative part f - is integrable. (See (8.4.9).)

8.1. Preliminaries

345

8.1. Preliminaries Given a norm-bounded sequence of elements f,,, in a Banach space E, it is often important to find in E an element cp in some sense asymptotically close to a subsequence of fn. Bounded subsets of reflexive Banach spaces are weakly sequentially compact (in fact, this is a characterization of reflexivity) so in Lp(SZ, F, p) for 1 < p < oo the weak limit of a subsequence of fn will do for W. Weak sequential compactness is not available in Ll (St, F, µ) without

extra assumptions (uniform integrability; see Section 2.3), so we have to settle for less than a weak limit. One procedure (8.1.4) is to consider elements of Li as members of Li*, i.e., finitely additive measures. Then by a theorem of Alaoglu, a subsequence of fn converges weak-star to an element 77 in Li*, and the maximal countably additive measure dominated by 77 will do for W. A related method

consists in taking Banach limits; see for example Krengel [1985], p. 135. First we discuss the truncated limit, a more elementary and transparent method of constructing V. Truncated limits

A weak unit in Ll is an element u E Li such that, for each f E Li , if u A f = 0, then f = 0. (This is the terminology used also in more general Banach lattices. See Remarks, below.) Any strictly positive integrable function u is a weak unit in L1; such a function exists since (1k, is or-finite.

Let (fn) be a sequence in Li with supra IIfnIIl = M < oo. A

function cp E Li is called a weak truncated limit of (fn) iff: for a weak unit u, the weak limit cpk = w lim fn A ku n

exists for every k E IN, and cpk T cp. It is easy to see that in this definition the choice of the weak unit u is irrelevant. We will write cp = WTL fn. n

If fn is not positive, WTL fn is defined as WTL fn - WTL f, , assuming these expressions exist. There is a compactness for weak truncated limits:

(8.1.1) Proposition. Let (fn) be a sequence in Ll with supra II A111 = M < co. Then there is a subsequence of (fn) that has a weak truncated limit cp, and IIcPIII < M.

Proof. It suffices to consider positive fn. Let u be a weak unit. For each k E IN, the sequence (fn A ku)nEIN is a sequence bounded by ku. Thus (fn A ku) is uniformly integrable, so it has a subsequence that converges weakly, say to cpk. By the diagonal procedure, we obtain one subsequence (fns) such that, for each k, the sequence fn3 A ku converges weakly to cpk as j -+ oo. Now the sequence c'k is increasing and bounded in norm by M,

346

Pointwise ergodic theorems

so its pointwise limit cp = limk cpk belongs to Li . Thus WTL3 fni = cp and IIVII1 0 whenever f > 0, then T is continuous with respect to the norm topology (and therefore also with respect to the weak topology).

(8.1.2) Lemma. Let fn ? 0 and gn > 0. If WTL fn = cp, WTL gn = y, and WTL(fn + gn) = 77, then cp + -y =,q.

Proof. First, (fn + 9n) A ku < (fn n ku) + (9n n ku), hence i < V + y. For the other direction,

(.fn + 9n) A 2ku > (.fn A ku) + (9n A ku),

which implies that q > V + y.

(8.1.3) Fatou's lemma. Let T be a positive operator on L1, and let f,, >0. If WTLfn=cpand WTLTfn=V, then TV 0, there exists m so large that T(ku) < mu + g, where g > 0 and II9II 1 < e. Then

T(cpk) = T(wlim(fn A ku)) = w lim T(fn A ku) n

< w lim (T fn A T(ku))

< w lim (T fn A (mu + g)) n

<wlim(TfnA(mu))+g n

yam+g oo) 1/1

U Ai 00

00

i=1

i=1

1b(Ai).

We say that 7P is countably superadditive. A charge 1/) is called a pure charge if it does not dominate any nontrivial

measure on A: if p is a measure and p < 1/i on A, then p vanishes on A. Similarly a pure supercharge is one that does not dominate any nontrivial charge. A partition of a set A is a collection of disjoint sets (in A) with union A.

(8.1.4) Theorem. Let 1/J be a supercharge defined on an algebra A of subsets of Q. Then: (1) V) admits a unique decomposition (8.1.4a)

1/I = VIM +

1b5,

where 1/Jm is a measure, 1/1c is a pure charge, and 1/J$ is a pure supercharge.

(2) The measure 4'm is given by (8.1.4b)

OM(A) = info

where info denotes the inf over all countable partitions {A1i A2, of A. The charge 1/1, is given by

}

0,: (A) = inff E(1/) -

(8.1.4c)

i

where inff denotes the inf over all finite partitions {A1, A2,

,

Ate,}

of A.

(3) If 1/I is a charge, then Os is 0. If 0 dominates a measure W, then 4Im also dominates gyp.

Pointwise ergodic theorems

348

Proof. Let 0 be a supercharge. We first show that bm defined by (8.1.4b) is a measure. Suppose {Al, A2i } is a countable partition of a set A. Given e > 0, choose for each i a countable partition {Ail, Ail, } of Ai such that b(Aik) C bm(Ai) + E2-Z. k

Then { Aik : i = 1, 2,. . . ; k = 1, 2,

} is a countable partition of A. There-

fore

1'm(A) < E i(Aik) < E [ibm(Ai) +,F2-'] _ E wm(Ai) + E. i

i,k

i

Since E is arbitrary, we conclude

bm(A)
0, choose a countable partition {B1, B2,. } of A such that E V,(Bk) < 'Im(A) +,F.

(8.1.4d)

k

Then

E

4'm ('4i)

i

LTV (Ai fl Bk) k

_E [(AinBk)] k

i

EV(Bk), k

since 0 is countably superadditive. Combining this with (8.1.4d), and taking into account the fact that e is arbitrary, we conclude t'm(A)

bm(Ai)

This completes the proof that Y)m is a measure. The proof that oc defined by (8.1.4c) is finitely additive is similar: countable partitions are replaced by finite partitions. The details are omitted. Next we show that Oc is a pure charge. Suppose µ is a measure on A and µ < c. Then, from (8.1.4c) we have u < 0 - Yam. For each set A E A,

bm(A) = info E i(Ai) ? info and therefore µ(A) = 0.

[Om(Ai) + iz(Ai)] = bm(A) + l.t(A),

8.1. Preliminaries

The proof that 'bs is a pure supercharge is similar. For the uniqueness, suppose that &i also has the decomposition

349

_

'Pm + cpc + cps, where Wm is a measure, cpc is a pure charge, and cps is a pure

supercharge. Note that the set function

0'(A) = inff TO. (Aj) is a charge dominated by 1/is, hence it vanishes. Similarly

cp'(A) = inff E cps(Ai)

vanishes. Therefore, applying the operation inff to both sides of the equation V m + Vc + 0s = cpm + Vc + cps, we obtain Y5m + Oc ='Pm + cpc. Similarly,

applying the operation info to this equation, we obtain Wm = 'm. Therefore also 4bc = Vc ands = cps. If 0 is a charge, then V),, = 0 by the uniqueness. Finally, suppose 0 dominates a measure cp. By the uniqueness, 'm = V. Applying the operation info to the inequality 0 > ', we have m > 0, then we write cp = TL fn iff, for some weak unit u, limn IIPk - (fn A ku)II1 -> 0 for all k, and cpk T V. For general fn, write TL fn = TL fn - TL fn .

Pointwise ergodic theorems

350

(8.1.6) (Weak and strong truncated limits 0 coincide for nonnegative functions.) Suppose f,,, > 0. Then WTL f,,, = 0 if and only if TL f,,, = 0. To prove this, integrate the bounded sequence fn A ku. A sequence (fn) will be called TL null if TL I fn I = 0. For this it suffices that w lim If,, I A u = 0. In v-finite measure spaces, stochastic convergence

is defined as convergence in measure on sets of finite measure. Thus in particular, in probability spaces, stochastic convergence is convergence in probability.

(8.1.7)

(TL convergence and stochastic convergence.) Let fn, o E Li.

Then the following are equivalent:

(1) TL fn = W. (2) fn - cp is TL null. (3) fn converges to cp stochastically.

(8.1.8) (Sharpening of Proposition (8.1.1).) If fn E Li , sup,, IlfnIii < oo, then there is a subsequence (fk,) of (fn) such that fk, = gn + hn, where gn, hn E Li , the sequence (gn) converges weakly, and the hn have disjoint supports (hence TL hn = 0). Remarks

The notion of truncated limits is particularly well suited for study of a large class of Banach lattices E, namely those satisfying the following two conditions:

(A) There is a weak unit; that is, an element u E E+ such that if f E E+ and f n u = 0, then f = 0. (B) Every norm bounded increasing sequence in E converges in norm. Assumptions equivalent with (B) are: (B') E is weakly sequentially complete; (B") E contains no subspace isomorphic to co. Condition (B) implies that E is order continuous, hence order intervals are compact and a weak unit exists if E is separable; thus the assumption (A) is not an important loss of generality (see Lindenstrauss & Tzafriri [1979]). Truncated limits exist in Banach lattices satisfying (A) and (B); the proof is the same as the Ll proof given here. The method of truncated limits was developed by Akcoglu & Sucheston [1978], [1983]. The decomposition (8.1.8) of Dacunha-Castelle & Schreiber [1974] could be

used instead in the real case, but this decomposition does not extend to Banach lattices satisfying (A) and (B); see Akcoglu & Sucheston [1984a]. Theorem (8.1.4), applied also in Chapter 4, is from Sucheston [1964]. For the purpose of the present chapter, the Yoshida & Hewitt [1953] decomposition in which zp is assumed to be a charge is sufficient.

8.2. Weak maximal inequalities There exist important ergodic results for operators acting on Li . One such result is E. Hopf's maximal ergodic theorem. We will first give a version of Hopf's maximal theorem for lattices.

8.2. Weak maximal inequalities

351

(8.2.1) Lemma. Let L be a linear space of measurable functions with values in (-oo, oo) such that L is a lattice under pointwise operations, let T be a positive linear operator on L, and let h E L. For N E IN, let n-1

hN =1max E Tih hN -T(h+). Proof. Since T is positive, it satisfies T(f V g) > T f V Tg. Thus T (h+) > TO V ThN

> OV (Th) V (Th+T2h)V ... V hN+1 - h > hN - h. For w E BN, we have hN(w) = h+ (w), so

h(w)1BN(w) = h(w) > hN(w) -T(h+)(w) = h+ (w) -T(hr+i)(w) For w

BN, we have h+ (w) = 0, so h(w) 1BN (w) = 0 > 0 - T (h+) (w) = h+ (w)

- T (h+) (w).

Let T be a positive operator on L1. Obviously T extends to the positive cone of positive measurable functions by

Tf=limTTfn 0, where

B,, = {supAN(f - A) > 01N

But B,,,) D If * > Al

{ f > A}, so f - A < 0 on B,,,, \ if * > Al. Therefore

f

(f - A) dµ > 0,

which yields the required inequality. Complements

(8.2.8) Theorem (8.2.2) remains true if the linearity of T is replaced by the assumptions TO

0,

T (f + 9) > T.f + Tg.

(8.2.9) (Point transformations.) A measurable transformation 9 maps Sl to SZ and is such that 9-1F C_ F. If also µ(9-'A) = µ(A) for each A E F, then 9 is called measure preserving or an endomorphism. An endomorphism

9 defines an operator T by T f = f o 9. Such an operator is Markovian: It is positive and preserves the integral, f f dµ = f f o 9 dµ. Remarks

Theorem (8.2.2) is due to E. Hopf [1954]. A great simplification of the proof is due to Garsia [1965].

An extension of Hopf's lemma to the non-linear setting was given by Lin & Wittmann [1991].

8.4. Hopf's decomposition

355

8.3. Hopf's decomposition We will next consider Hopf's decomposition of the measure space I into "conservative" and "dissipative" parts. Let T be a positive operator on L1. The corresponding potential operator Tp is defined by 00 n-1 lim E Ti f. Tpf = >2 Ti f = n-0o i=0

i=0

Pointwise convergence (possibly to oo) of the series holds at least for functions f _> 0. We will obtain the Hopf decomposition of the space f2 into the conservative part C and the dissipative part D, defined as follows. If

f E Li, then Tpf = 0 or oo on C; and Tpf < oo on D. We say that T is a conservative operator if fI = C, and T is a dissipative operator if Il = D. (8.3.1)

Theorem. Let T be a positive contraction on L1. Then S2

uniquely decomposes as a disjoint union C U D as above.

Proof. For nonnegative integrable functions f, g, define E(f, g) _ {Tpf = oo, Tpg < oo}.

Then Tp(f - Ag) = oo on E(f, g) for all A > 0. Applying Hopf's maximal ergodic theorem (8.2.2) to the function h = f - Ag, we obtain (since B,,. D E(f, g)):

0
0 a.e. Let C = {Tp fo = oo} and D = {Tp fo < oo}. By the argument

just given, with f = fo, we see that C is as required for the conservative part. To see that D is as required for the dissipative part, let h > 0 be integrable, and apply the previous discussion with f = h and g = fo. At each point where Tph = oo, we have either Tp fo = oo or Tp fo = 0. But

Tpfo> fo>O,soTphO,f =0 outside A}. A set A E Y is absorbing if T f E Li (A) for all f E Li (A). It will be proved that the conservative part C is absorbing. The absorbing subsets of C figure prominently in the identification of the limit in the ergodic theorem (see, for example, (8.5.3)). Therefore it will be useful to give several characterizations of this class. We will write C for the class of all absorbing subsets of the conservative part C. The first characterization of C is in terms of the potential operator Tp.

(8.4.2) Lemma. A subset A of C is absorbing if and only if it has the form {Tp f = oo} for some f E L'. The conservative part C is absorbing.

Proof. For f E Li , write Cf = {Tp f = oo}. Let A be an absorbing subset of C. Let f be an integrable function, 0 outside A and strictly positive inside A. Then Tpf is 0 outside A, since A is absorbing, and Tpf is positive inside A and hence oo since A C C. Therefore A = Cf. Conversely, suppose A = Cf. Let g E Li be 0 on A and strictly positive outside A. Since Tpf is finite outside A, we can write Il \ A as a disjoint union of sets Gi = fig < Tpf < (i+1)g} of finite measure. Now assume (for purposes of contradiction) that A is not absorbing. Then there is h E Li (A)

such that Th is not supported by A. Then for some i, the set G = Gi satisfies II1cThIl1 = a > 0. For each k E IN, we have II1GT(kh)II1 = ka.

Let f,,, be any sequence in Li such that fn increases to oo on A. Then limn (I (kh A fn) - khll1 = 0, so that

lnm IT((khAf,,)-kh)I

0,

and hence limn fG T(kh A fn) dµ - fG T(kh) du = 0, so limn fG T(kh A fn) dµ = ka and therefore limn fG Tf,, dp > ka. Thus limn fG Tfn dµ = oo.

8.4. The a-algebra of absorbing sets

357

We now obtain a contradiction by constructing a sequence f,,, such that this relation fails. We know that Tpf = 00 on A, so n-1

fn=ETif j=0

increases to oo on A. Then fn < Tpf and T fn < Tpf so we have fG T fn dp < (i + 1) fG g dp < oo. This completes the proof that A is absorbing.

Finally, C = C j for any strictly positive integrable function f, so C is absorbing.

The next characterization of C will involve the adjoint T* of the operator

T. Since T is an operator on L1, the adjoint T* is an operator on L,,,, characterized by the duality relation

fEL1i9EL..

I

Since we are assuming that T is a positive operator, it follows that T* is also a positive operator. We are also assuming that T is a contraction in

L1, so T* is a contraction in L. Thus T*1 < 1. Now the operator T* is monotonely continuous, that is, if fn T f E L., then T* fn T T* f. (See (8.4.8).) Therefore, T* (like T) extends uniquely to the convex cone of finite, nonnegative measurable functions, and the extension retains the monotone continuity (8.4.9). We will write TP for the potential operator defined from the adjoint operator T*.

(8.4.3) Lemma. Let A E C, and let h E L,,. be nonnegative on A. Then TPh has only values 0 and oo on A. In particular, TPh = 0 or o0 on C for every function h E LO. such that h > 0 on C. Proof. Using the duality relation we see that if A is absorbing for T, then SZ \ A is absorbing for T*. Therefore 1A T*h = 1A T*(h 1A), and by induction it follows that T*kh > 0 on A for all k E IN. If the lemma

fails, then there exists a positive number b and a set B C A such that 0 < TPh < b on B and 0 < µ(B) < oo. Then for each k we have 00 > b- µ(B)

J

T*Zh1Bdµ = JT*h.>Tu1B 00 > 0, i=k

i=0

which implies that T*kh = 0 on B. It follows that TPh = 0 on B, which is a contradiction.

Pointwise ergodic theorems

358

(8.4.4) Lemma. Let A E C and let g be a finite, positive, measurable function. IfT*g < g on A or T*g > g on A, then T*g = g on A.

Proof. Suppose T*g < g on A, and assume first that g E L+. Set h =

g - T*g. Then Ea T*ih = g - T*' g < g < oo. By Lemma (8.4.3), we O

have TPh = 0 on A, so h = 0 on A. That is, g = T*g on A. Now suppose g is unbounded and T*g < g on A. For a constant A, let g' = g A A. Then T*g' < T*g A T*A < g A A = g' on A. By the previous argument, T*g' = g' on A. If we let A T oo, we will obtain by the monotone continuity of T* that T*g = g. If T*g > g on A, proceed in a similar manner, using h = T*g' - g' and

the fact that T*l g' is bounded in L. The following theorem states the main results about C.

(8.4.5) Theorem. (1) The class C of absorbing subsets of C is a o-algebra of subsets of C.

(2) The class C is the class of all sets of the form C f = {Tp f = oo}, where f E Li . (3) The class C is the class of all subsets A of C such that T*1A = 1A on C.

(4) A nonnegative measurable function on C is C-measurable if and only if T*h = h on C. (5) A function h E L,, (C) is C-measurable if and only ifT*h = h on C. Proof. Let I denote the class of subsets A of C such that T*1A = 1A on C. First, T*1 < 1, so T*1C < 1C on C, hence by Lemma (8.4.4), we have T*1C = 1C on C. More generally, the inequality T*1A < 1A holds on A, so if A E C, we have T*1A = 1A on A. Let A E Z. If B = C \ A, then taking differences and using T * 1 C = 1C on C, we obtain that T * 1 B = 1 B on A. But T * 1 B < 1 B also holds on B, hence T*1B < 1B on C. Subtracting this from T*1C = 1C, we obtain T*1B = 1B on C. Thus I is closed under complementation. Thus C C Z. If A, B E Z, then on C we have 1A + 1B = T*(1A + 1B) = T*(1AnB + 1AUB) = T*1AnB +T*1AUB 1AnB + 1AUB = 1A + 1B So we have equality, and therefore T*lAnB = 1AnB and T*1AUB = 1AUB

on C. This shows that I is an algebra. The operator T* is monotonely continuous, so I is closed under increasing unions. Therefore I is a or-algebra. Suppose A E Z. Then also B = C \ A E Z. So T * 1 B = 1B. Now if f E Li (A), then

L Tfdµ f T*1B f dp=f 1B.f dp=0,

8.4. The o-algebra of absorbing sets

359

so T f is supported in A. Therefore A E C. We have shown that I C_ C, and hence that C = Z. Let H denote the convex cone of finite, positive, measurable functions

on C such that T*h = h on C. Observe that H is closed under infimum (and supremum). Indeed, T*(h A h') < T*h AT*h' = h A h' on C, so by Lemma (8.4.4), T*(h A h') = h A h' on C. Now we claim that a positive, finite, measurable function h belongs to H if and only if h is C-measurable. Indeed, if h is C-measurable, then it is the limit of an increasing sequence of lineax combinations of indicator functions of sets in C, so h E H by the

monotone continuity of T*. Conversely, since 1{h>a} is the limit of the increasing sequence

1An(h-a)+, we see that if h E H, then all sets of the form {h > a} are in C, so h is C-measurable. Finally, assume that h is in L,,,, (C). If h is C-measurable, then so are h+

and h-, so T*h = h on C by the preceding part. Conversely, if T*h = h on C, then T*(h+) = T*h +T*(h-) > T*h = h+ on the support of h+, hence T*(h+) _> h+ on C. Again by Lemma (8.4.4), it follows that T* (h+) = h+ on C. Thus h+ is C-measurable. Similarly, his C-measurable.

(8.4.6) Definition. Let T be a positive linear contraction on L1, and let A E C. The operator induced by A on T is the operator TA on L1(A) defined by

TAf = T(f 1A). The induced operator TA is conservative and Markovian (that is, it preserves integrals). The powers of TA are given by

TAkf =Tk(f .1A) The adjoint TA* defined on LE(A) satisfies TA*k(h) = T*k(h . 1A)

on A. The a-algebra of absorbing sets of TA is A fl C. All these statements follow from the preceding material, using the fact that A is absorbing. The operator T acts like the conditional expectation in that C-measuable functions may be factored out. Thus the appearance of EC in the limit theorems for T is not surprising.

Pointwise ergodic theorems

360

(8.4.7) Proposition. Let f E L1(C) and let h E L,,,, (C) be C-measurable. Then T(h Proof. Since T is monotonely continuous and linear, it is enough to con-

sider the case h = 1A, where A E C. Since A is absorbing, T(1A f) = 1A T(1A f). Now B = C \ A is also absorbing, so 0 = 1A T(1B f). Adding, we getT(1A f) = Complements

(8.4.8) (Monotone continuity) We write fn T f if fn is a sequence of functions, fn(w) < fn+1(w) for all n and almost all w, and f(w) = limn fn(w) for almost all w. A similar definition can be given for fn 1 f. A positive operator T is monotonely continuous on the space E of functions iff: fn T f E E implies T fn I T f . By considering the functions f - fn, we see that monotone continuity can by characterized by: fn 10 implies T f,, 10. Every positive operator on LP is monotonely continuous for 1 < p < oo, and every adjoint positive operator on L,,,, is monotonely continuous. The proofs are similar. Consider an adjoint operator T on Lam, say T = S*, where S is an operator on L1. If fn 10 in Lam, then T fn decreases to some nonnegative limit h. We must show h = 0 a.e. If g E L1, then

J

S9 - fndp=0.

This holds for all g E L1, so h = 0 a.e. (The relevant property of LP is "order continuity." See Lemma (9.1.1).) There is a converse in the case of Lam: If S: L,,,, - L,,. is monotonely continuous, then S = T* for some T : L1 -* L1.

(8.4.9) (Operator extension.) A positive operator defined on LP can be extended, preserving monotone continuity, to the convex cone of all nonnegative measurable functions (see also Neveu [1965a], p. 188). (8.4.10) (Continuity.) Every positive operator on a Banach lattice is automatically norm continuous. See (9.1.1), below (or Lindenstrauss & Tzafriri [1979], p. 2).

(8.4.11) (Point transformations.) Let T be the Markovian operator defined by an endomorphism 9 of a probability space (St,.F, µ). A set A is absorbing if and only if A = 9-'A a.e. So in this case, C is the a-algebra of invariant sets. (8.4.12) (Monotone continuity.) In an order-continuous Banach lattice E, if fn T f E E, then 11fn - f 11 -+ 0 (see Section 9.1 or Lindenstrauss & Tzafriri [1979], Proposition 1.a.8). Thus positive operators are monotonely continuous. For example, positive operators on LP spaces 1 < p < oo are monotonely continuous. Remarks

Study of the v-algebra C goes back to E. Hopf [1954].

8.5. Chacon-Ornstein theorem

361

8.5. The Chacon-Ornstein theorem (conservative case) This famous theorem asserts the convergence of ratios of the form

Rn=RX,9)_ En°TZf

i=o Tig

for f, g E L1, with g > 0. The case where T is conservative will be treated first. The nonconservative case will be done later (8.6.10) in the framework of superadditive processes. As is customary in ergodic proofs, we establish convergence for larger and larger classes of functions. If f is of the form s - Ts, where s E L1, then the numerator "telescopes," and we have Tns (f,9) = n-1s Ti n-1 Ti i=o The first term clearly converges to 0 on C. We will show that the second term converges to 0 on fl. (8.5.1) Lemma. Let s E Li , g E L1, g > 0. Then Tns En-1 Ti 9 i=o

9- =0 9

converges to 0 a.e.

Proof. Fix E > 0, let hn = Tns - E E? i= O Tig, ho = s, and An = {hn > 01-

Now ry = gµ is a finite measure equivalent top. We claim that En ry(An) < oo. To see this, first compute

f

f T(hn) dµ < f T(hn) dµ < f hn dµ = f hn dµ.

Thn dµ < n+1

o

S2

Note that Eg = Thn - hn+1i so that E

f

gdµ=f

Thndµ-f

hn+l dµ

fhndfh1 dµ. n}1

Thesum telescopes: 00

7(An)=> f gdµ< - f hodµ 0 a.e., and let ek E L1, k E IN, satisfy >k Ilek II1 < oo. Then limk sup. R,,(ek, g) = 0.

Proof. Write, as usual, -y = gp. Fix A > 0. Now apply Proposition (8.2.3). If sk = sup. Rf(Iek I, g), then A7{sk > A} a} g I

dry < fn IekI dp = IIekIIII

Therefore >k y{8k > Al < oo. By Borel-Cantelli, we have sk < A except for finitely many k, or lira supk sk < A. Since A > 0 was arbitrary, we have limksk = 0. Since IR.(ek,g)I < R, (Iek1,9), we have the required result. We will now assume that the operator T is conservative (C = Sl). Since < 1, hence T* 1 < 1. But T is conservative, so this means that T* 1 = 1; thus T preserves the integral, that is, T is Markovian. The a-algebra C is used to identify the limit of R,, (f, g). (See Section 2.3 for a discussion of conditional expectations on infinite measure spaces.) T is sub-Markovian, we have I I T* II

(8.5.3) Proposition. Let T be a conservative Markovian operator. Let f, g E L1 with g > 0 a.e. Let y = gp be a finite equivalent measure. Then the ratio Rn(f, g) converges a.e. to the finite limit

R(f,g) = E7 [i]. Proof. We may assume f > 0. If f is of the form h g, where h E L,,. and T*h = h, then by Proposition (8.4.7), we have Ta(h g) = h Tag, so clearly Rn (f, g) = h. But also, since h is C-measurable, E7 [ f g-1] = E7 [h] = h. If f is of the form s -Ts where s E L1, then by Lemma (8.5.2), we have R,, (f, g) -, 0 a.e. But also for A E C, we have fA(f /g) dry = f 1A f du = f 1A (s - Ts) dp = f (1A - T* 1A) sdp = 0, so that E7 [fg-1] = 0. Thus the theorem is true for all f in the linear space

Eg is dense in L1. By the Hahn-Banach theorem (5.1.2), it

suffices to prove: if k E L,, and f k f dp = 0 for all f E Eg, then k = 0 a.e. So suppose k is such a function. Then we have, for all s E L1,

J(k_T*k).sd/t=:Jk.(s_Ts)d/L=O, so that T*k = k. Thus we have k g E Eg, so 0 = f k k g dp, so that k = 0 since g > 0. This completes the proof that Eg is dense in L1.

8.5. Chacon-Ornstein theorem

363

Finally, let f be a general element of L1. Choose a sequence fk E E9 with Ek-1 II f - fkIIl < 00. Write ek = f - fk and 8k = sup,, Rf(Iek1,g). Consider the inequality I Rn ( f,

g) - R(f, g) I

I Rn (fk, g) - R(fk, g) + 8k + E

[I]

The first term on the right converges to 0 for each fixed k as n --+ oo because

fk E E9. The second term converges to 0 as k -> oo by Lemma (8.5.2). The third term converges to 0 as k --+ oo since

>2IEC, [

e

] d-y=IIIekIIl < 00.

The formulas for the conditional expectation from Section 2.3 may be applied to this limit R(f, g). For example, we have R(f, g) =

EC [f] Eµ [g]

whenever the conditional expectations with respect top exist, in particular if p is finite. Let us consider next what can be done if g E Li , but g = 0 on a set of positive measure. Then g p is no longer equivalent to p. But there is still some finite measure equivalent to p, say p = rp. The isomorphism between Ll (S2, p) and Ll (S2, .F, p) maps T to the operator Tp defined by

f E Li(p)

Tpf = r-1T(fr)

Clearly we have TPf = r-1Tk(fr), Rn(Tp)(f,g) = Rn(T)(fr,gr), andTp is conservative if T is. The o-algebra of absorbing sets is the same for T and Tp. The spaces L ,,.(p) and Li(p) are the same and the adjoint operators TP and T* are the same, because

fi. Tgdp = f(Tpf)

gdp

=f =fir. T*gdp

fi

T*gdp

Now if f, g E Ll and g > 0, then f /r, g/r E Ll (p), and the preceding results applied to the operator Tp give that

limR.(f,g) = n

EP

[gr-1]

364

Pointwise ergodic theorems

It follows from the convergence of Rn (f, g) (and can be checked directly) that the expression on the right does not depend on the finite equivalent measure p chosen. The case when g = 0 on a set of finite measure can now be discussed. It is senseless to consider the limit limn Rn (f, g) if Rn (f, g) is defined for no n because of 0 denominator. But the convergence can be proved on the set G where Rn(f, g) is defined for sufficiently large n. Notation: if B is a measurable set, we will write A E B fl C if A = B fl C for some C E C, or (equivalently) A C B and A E C.

(8.5.4) Proposition (conservative Chacon-Ornstein). Let T be conservative, f E L1, g E Li . Then the ratio

.

TZf

Rn(f,9)nO Ti9 i=O

converges to a finite limit R(f, g) a.e. on the set G = {Tpg > 0}. The limit R(f, g) is C-measurable, and satisfies fA g R(f, g) dµ = fA f du for all A E G fl C. If p = rµ is any probability measure equivalent to p, then R(f19) = EP [f r-1] EP [9r-1]

a.e. on G.

Proof. We claim first that the support F of EP [g] is G = {Tpg > 0}. By Theorem (8.4.5), G E C. Since g is supported in G, so is EP [g]. Thus F C G. Conversely, since G \ F E C, for each k,

0=f

G\F

EP [9] dp = f

G\F

Tkgdp,

so p(G \ F) = 0. By considering the induced operator TG as in Definition (8.4.6), we may assume G = Q. Now

Rn(f,9) =

B"(f,EP [9])

R. (9, EC [g])

By Proposition (8.5.3), the numerator converges to EP [fr-1] /EP [gr-1] and the denominator converges to 1.

The ratio R n (f, g) for f E L1, g > 0, still converges to a finite limit on the set {Tpg > 0}, if T is not necessarily conservative and g is not necessarily integrable. These generalizations are treated in the framework of superadditive processes in Section 8.6.

8.6. Superadditive processes

365

Complements

(8.5.5) (Point transformations.) Theorem (8.5.3), in the special case of an endomorphism of a probability space, with g = 1, is called Birkhoff's theorem, and is essentially the oldest pointwise ergodic result (G. D. Birkhoff [1931] ). The limit in Birkhoff's theorem is EC [ f ], where C is the a-algebra of invariant sets. Remarks

Theorem (8.5.4) is due to Chacon & Ornstein [1960]. The general (nonconservative) case will be proved below (8.6.10). The use of the Hahn-Banach theorem in the proof of the conservative case of the Chacon-Ornstein theorem originated with Neveu; see Neveu [1964] or Garsia [1970].

8.6. Superadditive processes Let T be a sub-Markovian operator. The sequence (sn) of measurable functions is called a superadditive process if (S1)

sk+n > sk + TkSn

k,n > 0

and

(S2)

'y = sup n

f sn dµ < oo. n

The number -y in (S2) is called the time constant of the process. A sequence (Sn) is called extended superadditive if (Si) is satisfied. A sequence (Sn) is

subadditive if (-sn) is superadditive; and additive if it is both superadditive and subadditive. Note that (Sn) is additive if and only if it has the form

n-1

Sn=ET=f i=0

for some function f.

If T is Markovian, i.e., it preserves the integral, then x,, = f sn dµ is a numerical superadditive sequence, i.e., Xn+k >_ xk + xn, and the time constant y = supnEIN n-1 f sn dµ is also given by ry = limn-,,,, n-1 f Sn dp (see 8.6.13).

The theory of processes subadditive with respect to an operator induced by a measure preserving point transformation was initiated by J.F.C. King-

man in 1968, who gave important applications to probability. Here, the operator superadditive theory is mainly developed because of the light it sheds on operator ergodic theorems. Mathematically, the superadditive and the subadditive cases are equivalent, but the first one is slightly simpler to treat for the following reason.

Pointwise ergodic theorems

366

An arbitrary superadditive process (sn) obviously dominates the additive process

the difference is a positive superadditive processes which can be studied in the pleasing context of positive operators acting on positive functions. By the same argument, if the additive theorem is known, we can restrict our attention to the positive superadditive processes. Dominants We will compare a positive superadditive process to an additive process that dominates it but barely. A dominant of a positive superadditive pro-

cess (sn) is an Li function 6 such E

O

Tib > sn for all n. An exact

dominant of a positive superadditive process (sn) with time constant ry is a dominant 6 such that f 6 dp = ry. We will show that such a 6 exists in some cases . We will use the functions M

WM =

7rt

(si - Tsi_1). i=1

(8.6.1) Lemma. Let cpn,, be as above. Then n-1

E TZco

>

1-

i=0

n-1

sn

m

for1 s1 +Tsi_1, so each term si -Tsi_1 is nonnegative. Next we have n-1

n-1

n

ETi(sj -Tsj-1) = ETZsj - ETzsj-1 i=0

i=0

i=1 n

_ (I -Tn)Sj + ETi(sj - sj_1), i=1

so that (remembering so = 0) n-1

rn

n

mET2cor = (I -Tn)1: sj+ETis,m. j=1

i=0

n-1 i=0

i=1

r.-n n-1 Si + Sn + E (Sn+i -Tnsi) + 1:(T2sn - TnSi+m-n). i=1

i=1

8.6. Superadditive processes

367

The first sum in the last line is nonnegative. Since sn+i - Tnsi > sn, the second sum is > (m - n)sn. Also, Ttsn-i 2! 0,

T''Sm - Tnsi+m-n = TZ(sm - Tn-Z8i+m-n) so the last sum is nonnegative. Therefore M n-1

E Ti'pm > [1 + (m - n)] sn. i=o

Now divide by m.

(8.6.2) Proposition. Suppose T is a Markovian operator, and (sn) is a positive superadditive process. Then (sn) admits an exact dominant.

Proof. For this proof, we use the weak truncated limits (see Section 8.1). Define cpn,, as before. Since T is Markovian,

f

Vmdµ

M 1 z-1

Y szdµ

-sz-1dµ

m 1

f

sm,dµ Ai > sn

(8.6.2a)

n > 1.

i=o

By the Fatou property of the weak truncated limits (T WTL < WTL T, see 8.1.3), we have TAi-1 < A,, and, more generally, TnA1 < Ai+n. Hence we can write

Ai = (Ai-TAi_1)+T(Ai_1 -TAi_2)+...+Ti-Kill -TAo)+TZAo, with all the summands positive. Therefore 7 >_

f

Ai dp

= f [(Ai - TAi-1) + (Ai-1 - TAi-2) + ... + (A1 - Tao) + A0] dµ. Define 00

b = AO + E(A1 - TAi-1) (8.6.2b)

i=1 cc

_ E(A. - TA,). i=o

Pointwise ergodic theorems

368

Then f b dy < y, and n-1

00

Tib = (I - Tn)

Ai,

j=0

i=o 00

00

>1: Ai

- EAi

i=0

i=n

n-1

=EAi>Sn ii=0

by (8.6.2a). This shows that n f 6 du > f sn dµ for each n, hence f 6 dp > -y. Thus f Sd p = ry. This shows that 6 is an exact dominant. A positive superadditive process (shifted) can be approximated also from below by additive processes. Note that a positive superadditive process is increasing: Sk+n > Sk + Tksn > Sk

(8.6.3) Lemma. Let 8k be a positive superadditive process, and let ak = Sk/k. Then n-1

E TZak < Sn+k-1 i=0

for alln>0. Proof. We have n-1

n-1

Tiak = E Tisk

k

i=0

i=0

n-1 < E (Sk+i - Si) i=0

n+k-1

n-1

E Si - E si < ksn+k-1, i=0

ii=0

where the last inequality follows from the fact that sn is increasing. Superadditive ratio theorem

(8.6.4) Theorem. Let T be a conservative operator, and let sn be a positive superadditive process with an exact dominant 6. Let Cb = {E°_0 Ti6 > 0}, and define v(A) = fA S dµ. Then (8.6.4a)

lim n

s

n-1 i=0

1 ib

on C6

8.6. Superadditive processes

369

and

J sn dµ = a(A) n n1 JA sn dµ = sup n n1 A

(8.6.4b)

for A E C6 fl C.

lim

Proof. Let f,g E Li . Set C. = {EO°0Tig > 0}. From Proposition 8.5.4 we have that lim Rn(f, g) = R(f, g) where R(f, g) is C-measurable and

jR(f,g).gd=ffdL

(8.6.4c)

for A E C9 fl C. Now T* 1A = 1A, so the right-hand side does not change if f is replaced by Tk f . Therefore, for each k, we have R(f, g) = R(Tk f, g). Furthermore, CTk9 = C9 and R(f, g) = R(f, tkg). So

E TZf lm . 0 TZf n+k-1 0 T gi= n E n+k-1 i Tg

R (f, (f, g) =

i=k

So we have

n kTi R(f, g) = lim

(8.6.4d)

n

En-_ol 7' if g

i=01

By Lemma 8.6.3 and the definition of a dominant, if ak = Sk/k, then n-k

n-1

E Tiak < sn
s' for all n and, for all A E C,

8'dµ=o'(A), JA

where a'(A) is the (finite or infinite) limit

o'(A)=limnJ s' dµ=supJ s',,dµ. nnA A The existence of extended exact dominants can be established by applying the results about exact dominants to processes of the form 1AS'n where A E C and o'(A) < oo. (See also the proof of Theorem 8.6.7, below.) We now show that an extended superadditive process divided by n can be integrated term by term. (Of course, this is false without superadditivity.)

(8.6.6) Lemma. Let

Yin

be a sequence of measurable functions withn

integrable. Suppose (8.6.6a)

fork,n>0.

1'k + ,bn C V)k+n

If E is a conditional expectation operator, or an operator fA, then

-oo < lim -Eon = E ( lim 1 in) < oo.

\nn

nn

on/n. Therefore Proof. We have by induction using (8.6.6a) that Y)1 2Gn/n is bounded below by the integrable function -iii , so that Fatou's lemma applies. Using (8.6.13), we have

-oo < E01 < E (lim 1 On) n

< lim E

/

=E (supnon)

= E I lim 1 On) < oo. \\\ n n /

0

We will now consider the general non-conservative case. Here is the main

theorem. Recall: C is the conservative part, D the dissipative part; T is Markovian if it preserves the integral.

8.6. Superadditive processes

371

(8.6.7) Theorem (Akcoglu-Sucheston). Let T be a sub-Markovian operator, s,, a positive superadditive process, and s' a positive extended superadditive process. Let E = {sup,, sn > 0}. Then lim sn/sn = h exists a.e. on c fl E and is C-measurable. Let 6 [respectively, 6'] be an exact [extended exact] dominant of the process sn 1c [sn 1C] with respect to the Markovian operator TC. Let p be a probability measure equivalent to p, say p = rp. Set s = sup 1 EC [sn 1c/r]

nn

s = sup -EC [sn Ic/r].

nn

The following limits exist a.e. on C:

linm nEp [sn Ic/r] = s = EC [6/r] < oo, p

(8.6.7a)

linm

nEp [sn lc /r] = s' = EC [6'/r] < oo. p

Also, lira sn/sn = s/s' a.e. on C fl E. If either T is Markovian or sn is additive on D, then lira sn/s'n = lira I sn/ lim T s', a.e. on D fl E. Proof. To simplify the notation, we will assume that p is a probability and r = 1. To obtain the general case, replace p by p and Ec by EP [ r-1]. Since sn and sn are increasing sequences, lim sn/s' = lira T sn/ lim T s', exist a.e. on D fl E if lim sn < oo on D. This is true in the Markovian case because of the existence of a dominant for sn (see Proposition 8.6.2). If sn is additive on D, then it is of the form EZO Ti6 with 6 integrable, hence lim sn < oo on D. Now consider C. If (sn) is (extended) superadditive, then so is (sn 1A) for each absorbing set A, both with respect to T and TA, because sk+n 1A >- Sk 1A + (Tksn) 1A > sk 1A +Tk(sn 1A)

Positive linear operators on an LP space (or an Orlicz space) can be uniquely

extended to the set L of all measurable functions f from 1 to (-oo, oo] such that f - is integrable (8.4.9). The extension preserves their monotone continuity. This is, in particular, true for the sub-Markovian operator T, and for the conditional expectation EC []. Since the restriction to a set A E C of an (extended) superadditive process is (extended) superadditive, choosing A = C, we may consider the operator Tc instead of T, and thus assume that S2 = C. If A E C, then T* 1A = 1A implies that fA f du = fA Tf dp and also EC [f] = EC [T f ]. Therefore, applying Lemma 8.6.6, first with E = fA, then with E = EC, we obtain, for each A E C, (8.6.7b) A

s'ndp=sup1J sndp=Q(A)

(i+--) m

s2,

hence ''o + T**?o > s2µ and i7o+iro+T**(170+fro) _,70+.7ro+Tr7o+771+7r1 > s2µ.

But 7r1 and ir2 are pure charges, and a sum of pure charges is a pure charge, and so does not dominate any measure; so we have rlo +Tr7o +77l >- s2µ.

In general, given 7rn, let 71n+1 = M(T**7rn), and 7rn+l = T**7rn - 11n+l Finally, set 77 = E°_0 77i. Then n-1

n-2

n-1

T277o+ETiq1+...+r7n-1>snµ,

Ti17

i=0

i=0

i=o

It follows that i7(1) = ry, so that S = dr7/dµ is an exact dominant. (3) (2). Now assume there is a dominant 6. Then 0
0. Then

Rn(f,g)_ En-

i=o i=01

T2g

converges to a finite limit h a.e. on the set E = {Tpg > 0}. Let p = rµ be a probability measure equivalent to µr. The limit h is equal to E0 [Hf

EP [ 2]

376

Pointwise ergodic theorems

a.e. on c fl E and equal to

roo

Mo

Tif

F.i°°

=0Tig

a.e. on D n E. Proof. Use 8.6.7 and 8.6.9 with sn = E i= O Ti f and s', = E i= O Tig. Next we prove the Dunford-Schwartz Theorem for positive operators. That the supremum of An f in the Dunford-Schwartz theorem is finite, follows also from 8.2.6, above.

(8.6.11) Theorem (Dunford-Schwartz). Let T be a sub-Markovian operator which also satisfies T1 < 1. If f E Ro , then n-1

An.f=nETif i=0

converges a.e. to a finite limit h. On the conservative part C,

h= On the dissipative part D,

Tif h = E000 =0 Til Ei0o

Proof. For each f E Ro, and each constant A > 0, there is a function f A E L1 such that If - f AI < A. Since IAnAI < A and A is arbitrarily small, we may assume f E L1.

Now sn = E o TI f is additive, while s' = n 1 is extended superadditive, because k + n > k + T(n 1). W e have s' = limn(1/n)EP [1 n1r-1] _ Ep [1cr-1]. (See also the identification of the limit in Theorem 8.6.12, below.)

Our last ergodic theorem is Chacon's theorem for positive operators. A sequence (pi) of positive measurable functions will be called admissible if Tpi < pi+1 holds for all i > 0. For example, (pi) is admissible if pi = Tig for some positive g. Also, if T1 < 1, then the sequence pi = 1 is admissible. These two examples show that both the Chacon-Ornstein Theorem 8.6.10 and the Dunford-Schwartz Theorem 8.6.11 are consequences of the following.

8.6. Superadditive processes

377

(8.6.12) Theorem (Chacon). Let T be a sub-Markovian operator, let f E L'1, and let (pi) be admissible. Then

En-1 Ti i=0

En-1

f

i=0 pi

converges to a finite limit h a.e. on the set E S' = limn Ep [lcpnr-1]. The limit h is equal to

> 0}. Let

Ep [H,] s'

a.e. on the set c n E, and equal to Elm i=o

Tf

a.e. on D n E.

Proof. Since Tpi < pi+1 for each i, we have by induction Tkpi < Pi+k Thus

n+k-1

E

i=k

nr-1

pi > T k ? pi. i=0

Hence s'n = En-I pi is extended superadditive. Set sn = E2

Ti f . Now O Theorems 8.6.7 and 8.6.9 may be applied. The result follows, except that s' is the limit of the Cesaro averages of EC [lcpn], rather than lim Ec [lcpn] itself. (As before, we assume µ is a probability measure and r = 1.) However, Ec [lcpn] = EC [Tcpn] < EC [lcpn+l], so that the sequence EC [lcpn] is increasing, and the two expressions for s' coincide. The observation that Ep [lcpnr-1] must be increasing if Pn is admissible indicates that superadditive processes are much more general than processes that are sums of admissible sequences. Complements

(8.6.13) (Numerical superadditive sequences.) Suppose Xn E (-0o,00] satisfies Xk+n > xk + xn. Then xn

lim Xn n n n-.oo n = sup

.

To see this, write -y = supra xn/n (finite or infinite). Fix a positive integer d. Each n can be written with quotient kn and remainder rn, where n = knd + rn and 1 < rn < d. Then xn > xr . But Xk,d > knXd, so xn

n

(knd) xd/d + xr,. n

Pointwise ergodic theorems

378

Now knd/n -> 1 and x,.,, /n -* 0, so we have

limn inf n > d . This is true for all d, so 1im inf xn/n > y. The inequality lim sup xn/n < y is clear.

(8.6.14) The operator H used in 8.6.9 may be generalized as follows. If A E F, define the operator IA by IA(f) = 1A f . Define HA by

HA f = lA j:(TIo\A)i f. i=0

If A E C, then Proposition 8.6.9 remains valid with the operator HA in place of H. However, the operator HA is sub-Markovian even if A is not absorbing. Hint: Define HA(') in a manner analogous to H(s), and show by induction on n that H(An)

= (IA+TIo\A)n - (TIO\A)n.

Therefore H,(q) is dominated by powers of the positive contraction

IA + TIo\A.

(8.6.15) (Convergence of Cesaro averages An f .) Suppose there is a positive g such that {g > 0} = C and Tg < g on C. Then (1/n) En-O T' f converges a.e. on 12 for each f E Li . Indeed, the process s' = long is extended superadditive with respect to the operator Tc, so that the ratio Ti f /ng converges a.e. on C. But on D, it is clear that rZ o T' f /n o converges to 0. (8.6.16) For f E Li , set s,, = lim inf,. Ei+T-1 Ti f . Then sn is superadditive, so Theorem 8.6.7 is applicable. This is also true if lim inf is replaced by inf .

(8.6.17) (Subadditive theorem for measure preserving point transformations.) An example of a positive subadditive process to which Theorem 8.6.7 can be applied is 'n-1

sn = max

e;=±1

Ueif 00

I

i=0

where f is a Bochner integrable function with values in a Banach space, 0 is a measure-preserving point transformation on a v-finite measure space, and the max is taken over all measurable choices ei(w) of signs +1, -1.

8.6. Superadditive processes

379

(8.6.18) (More on the superadditive theorem for measure-preserving point transformations.) Let 9 be a conservative measure preserving point trans-

formation and p be a number with 0 < p < 1. Let f be a nonnegative measurable function such that f f P dµ < oo. Then, for any positive function g, r -1 E:=0 fo9 lim Ei=o _1 =0 z

n

g ° 9i

almost everywhere on {Tpg > 0}. (Akcoglu & Sucheston [1984].) If the measure i is finite, g = 1, and f o 9i is an independent sequence, then this result is due to Marcinkiewicz. (See also 6.1.15.) The ergodic result remains true if the pth power is replaced by a positive subadditive function

(8.6.19) (Markov kernels.) A Markov transition kernel is a function P(w, A), such that: (1) P(w, ) is a probability measure defined on .1 for each fixed w E ft (2) P( - , A) is a measurable function on ci for each fixed A E F. The kernel P is null-preserving if u(A) = 0 implies P(w, A) = 0 for almost all w. Such a kernel P defines a Markovian operator T via the RadonNikodym isomorphism as follows: If cp is a measure on.F, define the measure Tcp by

f Tcp(A) =

J

P(w, A) cp(dw).

Since P is null-preserving, if cp is absolutely continuous, so is Tcp, so T defines an operator from L1 to L1. The adjoint operator T* on L,,,, satisfies

T*h(w) = Jh(y)P(w,dy). An important particular case is when P(w, A) = 1e-1(A) (w), where 9 is a measurable invertible point transformation that maps null sets on null sets. Then the Chacon-Ornstein theorem is essentially the ergodic theorem of W. Hurewicz [1944]. The connections between ergodic theory and Markov processes are studied in Revuz [1974].

(8.6.20) Recall that a sub-Markovian operator is a positive linear contraction on L1. There is an example, due to Chacon [1964], of a subMarkovian conservative (Il = C) operator T such that the Cesaro averages A,, f = (1/n) En-O Ti f diverge a.e. for each non null f E Li . This example also shows that there need not exist a strictly positive function g such that Tg < g, since the existence of such a g implies convergence of A" f (see 8.6.15). In fact, in Chacon's example, T is generated by an invertible non-singular (preserving null sets) point transformation 9 by T`o=9-'co,

co=fµ,

380

Pointwise ergodic theorems

which shows that Cesaro convergence may fail also in the setting of the Hurewicz theorem. In the Hurewicz setting, the nonexistence of a g such

that g > 0 and Tg < g (equivalently, Tg = g) also provides a negative solution to the famous problem of the existence of a "o-finite equivalent invariant measure"- this name is given to -y = gµ-earlier resolved by D. S. Ornstein [1960].

Even if An f converges to a finite limit, as is the case if TI _< 1, if p is infinite this limit is often zero (8.6.21), so that the Chacon-Ornstein theorem remains of interest, informing about the relative behavior of sums of iterates of two functions f and g such that lim A,,, f = lim An,g = 0. (8.6.21) (The ergodic case.) A sub-Markovian conservative operator T is called ergodic if the a-algebra C is trivial, that is, for every element A E C, either p(A) = 0 or p(1l \ A) = 0. If T is ergodic, f E Li , and g > 0, then the limit R(f, g) in the Chacon-Ornstein theorem is E [f] /E [g] a.e. on D. Indeed, we observed (following (8.5.3)) that for integrable g the limit in the conservative case is EC [f] /Ec [g]. The case of general g > 0 can be reduced to the integrable case by taking integrable positive gj's with gj T g and observing that E [gj] T oo and E [f] /E [gj] > E [f] /E [g].

If T is ergodic, T1 < 1, p(Q) = oo and f E Li , then the limit in the Ti f < R(f, 1) = 0. Dunford-Schwartz theorem (8.6.11) is lim (1/n) o (8.6.22) (The ratio theorem if T is not a contraction.) If T is a positive L1 operator that is power-bounded (i.e., such that sup, IITnII1 < oo), then the space S2 uniquely decomposes into parts Y and Z characterized as follows: if f E Li (Y), then liminf IITn f 111 > 0; if f E Li (Z), then JITn f X11 -+ 0. If f E Li and g > 0, then the limit R(f, g) exists a.e. on Y; in general it fails to exist on Z. (This is proved in Sucheston [1967]. Further results about the decomposition Y + Z were obtained by A. Ionescu Tulcea & M. Moretz [1969]; and by Y. Derriennic & M. Lin [1973].) Remarks

The pioneering paper of E. Hopf [1954] gave the first Ll operator theorem: Theorem (8.6.11) under the assumption that T1 = 1. Theorem (8.6.10) was conjectured by Hopf and proved by Chacon & Ornstein [1960]. The identification of the limit is due independently to Neveu [1961] and Chacon [1962]. The Chacon-Ornstein theorem for measure-preserving transformations is due to E. Hopf [1937]. The identification of the limit is facilitated by the fact that not only C, but also D is absorbing (J. Feldman [1962]). Theorem (8.6.7) and the derivation from it of the general Chacon-Ornstein and Chacon theorems is from Akcoglu & Sucheston [1978]. Theorem (8.6.8) is from Brunel & Sucheston [1979]. The identification (8.6.9) of

the operator H f as an exact dominant is new. The identification of the limit in Chacon's theorem (8.6.12) is due to U. Krengel [1985], p. 130. The proof via (8.6.7) and (8.6.9) of various ergodic results including Kingman's

theorem and Chacon's theorem together with identification of the limit is by far the shortest. However, other approaches may provide additional information. We mention two powerful methods, both presented in detail in Krengel [1985]. The filling scheme, the original method of Chacon-Ornstein, was widely applied in ergodic theory and probability. The maximal lemma of Brunel [1963] provides a proof of

8.6. Superadditive processes

381

Chacon-Ornstein that connects with potential theory; it was applied to identify the limit by P. A. Meyer [1965]. The finite measure case of the superadditive theorem for measure-preserving point transformations is due to the pioneering paper of J. F. C. Kingman [1968].

There exist striking applications, in particular to percolation theory and to the limiting behavior of random matrices (Kingman [1973] and [1976], Kesten [1982]).

9

Multiparameter processes

We present here* a unified approach to most of the multiparameter martingale and ergodic theory. In one parameter, the existence of common formulations and proofs is a well known old problem; see e.g. J. L. Doob [1953], p. 342. It has been known that the passage from weak to strong maximal inequalities can be done by a general argument applicable to harmonic analysis, ergodic theory, and martingale theory. In this book a very general such approach is presented in Chapter 3, involving Orlicz spaces and their hearts. There exists also a simple unified (martingales + ergodic theorems) passage from one to many parameters using no multiparameter maximal theorems, based on a general argument valid for order-continuous Banach lattices. This approach gives a unified short proof of many known theorems, namely multiparameter versions of theorems of Doob (Cairoli's theorem [1970] in stronger form, not assuming independence), theorems of Dunford & Schwartz [1956] and Fava [1972], the multiparameter pointtransformation case having been earlier proved by Zygmund [1951] and Dunford [1951]. We also obtain multiparameter versions of theorems of Akcoglu [1975], Stein [1961], Rota [1962]. For the Banach lattice argument, the order continuity is needed, which means that the L log" L spaces are not acceptable if the measure is infinite: they fail this property and have to be replaced by their hearts, subspaces H., which are closures of simple integrable functions (see Chapter 2). We will first develop in detail the "multiparameter principle" (Theorem (9.1.3)) that allows the reduction of multiparameter convergence problems to one parameter. There is also a one-sided version of this result, useful to prove "demiconvergence" in many parameters. Also a Banach-valued version of the convergence principle is presented, for operators that have positive dominants (Theorem (9.1.5)); many simple operators are in this class, including point transformations and conditional expectations. As an application, in Section 9.2, we deduce theorems about convergence of multiparameter Cesaro averages of operators. Next, a version of the multiparameter superadditive ratio theorem is obtained (Theorem (9.3.3)); this contains a multiparameter Chacon-Ornstein theorem. Finally, we consider *Parts of this chapter are taken from "A principle for almost everywhere convergence of multiparameter processes," by L. Sucheston and L. Szab6, pages 253-273 in Almost Everywhere Convergence II, A. Bellow and R. Jones, editors, Copyright © 1991 by Academic Press, Inc. Used by permission of Academic Press.

9.1. A multiparameter convergence principle

383

martingales. It has been known that less than the independence assumptions of Cairoli [1970] suffice for convergence theory; if a martingale is also a "block martingale" then the independence may be dispensed with (Theorem (9.4.5)). Conversely, in the presence of independence (or even properly

defined conditional independence-the condition sometimes called F4 or commutation), every martingale is a block martingale (Proposition (9.4.3)). Here we reduce the case of block martingale to successive applications of the conditional expectation operator, which allows the use of Theorem (9.1.3) and simplifies the proofs. Block martingale theorems imply strong laws of large numbers for two parameters (9.4.8).

9.1. A multiparameter convergence principle Let E be a sigma-complete Banach lattice; that is, a Banach lattice such that every order-bounded sequence has a least upper bound in E (see Lindenstrauss & Tzafriri [1979]). Recall that E is said to have an ordercontinuous norm if for every net (equivalently, sequence) (fi), fi 10 implies 111211

Let F C E. A map T: F -+ E is increasing, if f< g implies

T f < Tg; positive if f > 0 implies T f > 0; linear if T (a f +/3g) = aT f +,3Tg for any a,,3 E IR; positively homogeneous if I T (a f) I = a I T f I for each f E F,

a E IR+; subadditive if T (f + g) e > 0. Choose a sequence (in,) of indices such that 00

E 2nllfi II < oo and inf IITfi II > E. n=1 Set

n

9n = E 2kfk k=1

Since F is closed, gn T g E F+, and for every n, Tg > T9n ? T(2 n fin) = 2nT(fi ), hence

IIT9II >_ 2

ns.

Multiparameter processes

384

This is a contradiction, therefore T must be continuous at 0. If fi 10 then I I fi II 10, because E is order continuous. Therefore II T fi II 10. Also Tfi . g for some g E E+, and necessarily II9II mj, so Vufk > Vumj. Thus inf Vufk >_ Vumj

k> j

Therefore, for each i,

inf inf Vu fk > inf Vumj. u>i k> j

u>i

Letting i -> oo yields

liminf inf V fk > liminf Vimj > Vumj. i

k>j

i

For each j,

lim inf inf Vi f k < lim inf Vi f j . i

k>j

i,9

Therefore

liminfV fj > Vumj. i,j

Now the net mj increases to f,,,, E F+. The operator V,,. is monotonely continuous by Lemma (9.1.1(ii)). Hence Vumj T V3 fem. It follows that liminfV fj > Voofco. i,j

(ii) Necessarily foo E F+. For each j E J, let Mj = supk>j fk. Applying Fatou's Lemma as before, we have for each u, suPVufk j

Therefore, for each i, sup supVu fk < sup VuMj. u>i u>i k>j

Letting i -+ oo yields lim sup sup V f k < lim sup V Mj = Vo. Mj i i k>j

Multiparameter processes

386

For each j, limsupsupV fk >_ limsupV fj. i

i,j

k>j

Therefore

limsupVifj m II fj - fooll, so 8m E Rk_1(IR) C_ Ro(IR). Then lim sup 11 Vi f j- Voo foo 11

i,j

limsup llV(fj - foo)Il+limsuplluifm - Voofooll i,3

i

= limsup II v (fj - W11 i,j

< lim sup Villfj - fool) i, j

< lim sup Vism = V.. )sm. i

But sm -+ 0 and Ro(IR) has order-continuous norm (2.1.14), so Voosm + 0.

We now consider several parameters at the same time. As in Theorem (9.1.3), let Ii, I2i , Id be directed sets with countable cofinal subsets, and let I = Ii x 12 x . . . x Id be the product.

(9.1.5) Theorem. F o r i = 1, 2, .. , dand j E J, let T(i, j) be an operator on Lmax(E), positively dominated by T(i, j). Suppose

(a) For each f E Ro(E), the limits limj T(i, j) f = T(i, oo) f E Ro(E) and limj T(i, j)II f II E Ro(IR) exist. (b) For each f E Ri(E), supj T(i, j) 11 f II E Ri_1(IR).

Then for each f E Rd(E), lim T(1, t1)T(2, t2) . .T(d, td) f = T(1, oo)T(2, oo)

.

. T(d, oo) f

exists as the indices ti converge to infinity independently.

Proof. This is proved by induction on d using Lemma (9.1.4). For d = 2,

choose f E Ri(E), fj = T(2, j) f which converges to T(2, oo) f = fm, and V = T(1,i). For the general induction step, assume that the theorem holds for d parameters, let f E Rd+1(E), f j = T (d + 1, j) f , foo =

Multiparameter processes

390

T(d + 1, oo) f, V = T(1, i1) T(d, id). It is easy to see that the operator T(1, i1) T(d, id) is a positive dominant of Vi Remarks

The multiparameter convergence principle [Theorem (9.1.3(iii))], with most applications, is from Sucheston [1983]. The one-sided results [(i) and (ii) on demiconvergence] were developed in Millet & Sucheston [1989] and in Sucheston & Szabo [1991]. Theorem (9.1.5) was stated in FYangos & Sucheston [1986]. Demiconvergence in martingale theory was introduced in Edgar & Sucheston [1981], and further studied by Millet & Sucheston [1983].

9.2. Multiparameter Cesaro averages of operators In 1951, articles of A. Zygmund and N. Dunford appeared under the same title and in the same volume of Acta Sci. Math. (Szeged), proving multiparameter convergence theorems for noncommuting point transforma-

tions. Zygmund assumed f E Llogd-1 L of a probability space. Dunford allowed a v-finite measure but restricted f to Lp, p > 1. The first obvious challenge was to find the common generalization of these two settings. The spaces Rk introduced by Fava [1972] fulfilled this role (also in the more general Dunford-Schwartz operator context). However, the theorem of Zygmund, Dunford, and Fava still appeared as a difficult result, depending on a multiparameter maximal theorem: see e.g. Krengel [1985], pp. 196-201. Theorem (9.1.3) reduces the theorem to the one-parameter theory that had been known earlier (cf. N. Wiener [1939]). We first recall some of the results from Chapter 2 about function spaces and maximal inequalities. A finite Orlicz function (2.1.1) is an increasing convex function [0, oo) --> [0, oo), satisfying 0 and -P(u) > 0 for

some u. Such a 4 is differentiable a.e.; the derivative cp, defined a.e. by 4D(u) = fe cp(x) dx, will be assumed left-continuous. Often we assume in addition that -t(u)/u -> oo, which happens if and only if cp is unbounded. Then there is a "left-continuous generalized inverse" 0 of cp, defined by 1/i(y) =inf { x E (0, oo) : cp(x) > y }

.

The function 0 is a derivative of an Orlicz function 4', called the conjugate of &. We are also interested in the function t;, defined by e(u) = ucp(u) - 4(u) = T (cp(u)). It is left-continuous; it may or may not be an Orlicz function.

Let (St, .F, i) be a measure space. The Orlicz modular is the function

M4 (f) = f (I f I) dµ. The Luxemburg norm of a measurable function f is

II.fIIs=inf{a>0:Ms(f/a) c'\} < If

f dµ. >a}

Iff E Rk then g E Rk_1. This lemma, together with Theorem (8.2.6) shows:

(9.2.2) Lemma. Let T be a positive linear operator on L1 + Lam, and let A,,,(T) = (1/n)(T° + T1 + + Tn-1). Assume that IITII1 < 1 and supra IIAn(T)II. = c < oo. Iff E Rk then sup An(T) f = g E Rk_1

Suppose that T is a positive contraction of L1 and L. The oneparameter averages A,,(T) f converge a.e. for f c R0 by (8.6.11). Thus we have all the elements needed for the application of Theorem (9.1.3), with L(O) = L(1) = R0i L(i) = Ri_1 for i _> 2, and T(i, j) = Aj(Ti). This yields:

(9.2.3) Theorem. Let Ti be positive operators defined on L1 + L,,. such that IITII1 0, AP{Y > A} < E [1{y>-,\)X] If X E L logk L then Y E L logk-1 L. Proof. Since Xt1 converges in L1, it is uniformly integrable. Therefore we may pass to the limit under the integral sign in (6.1.5), with P1 = P. This proves the first assertion. The second assertion follows from (3.1.9(b)). Recall ((1.3.1) and (1.4.3)) that Li-bounded martingales (and submartingales) indexed by directed sets converge stochastically; and in the presence of uniform integrability converge in L1 norm. In this section we will consider martingales in d dimensions. We will see that a "block martingale" converges a.s. if it is properly bounded. A particular case is the convergence under condition (F4). The proof is by reducing block martingales to consecutive applications of conditional expectations and applying Theorem (9.1.3). Fix a positive integer d. The directed set I will be INd = IN x IN . . x IN, with d factors. The ordering on I is defined as usual: ifs = (s1i s2, , Sd)

and t = (t1, t2i , td), then s < t if s;, < ti for all i. Let (.Ft)tEJ be a stochastic basis. For integers i, j with 1 < i < j < d, write

for the a-algebra obtained by lumping together the or algebras on all the axes except for axes numbered from i to j. More precisely, ""i is the or-algebra generated by all T, with sk = tk for i < k < j and Sk arbitrary for other k. Of course the containment relations remain, so that for fixed i and j, the family ( ""j)tEI is again a stochastic basis. When i = j, we will sometimes write Tt for JTt'. Denote the conditional expectation Ewe

by E3and Ec8 by E8.

Since the index set I = INd is a directed set, the definition of Chapter 4 applies: a process (Xt)tE1 is a martingale [submartingale] if X. = E3Xt [X8 < EBXt] whenever s < t. Now I has additional structure, so other variants are possible. If k E IN, 1 < k < d, then we will say that the process (Xt) is a block k-martingale [block k-submartingale] if

El kXt = X (81,... ,8k,tk}1i... ,td) 9

Ir

E9 kXt > X81,...

1

,8k,tk}1,... std) J .

L

whenever s < t. An integrable process is a block martingale [block submartingale] if it is a block k-martingale [block k-submartingale] for all k < d. Block k-martingales should not be confused with "k-martingales," defined by E8 Xt

v

(tl ... ,tk-1>8k tk+1 ... ,td)

Block martingales may be characterized in terms of "factorization" of conditional expectations:

Multiparameter processes

398

(9.4.2) Proposition. (1) Let (Xt) be a uniformly integrable martingale. Write X = slim Xt. Then the following are equivalent: (a) Xt = EtX is a block martingale.

(b) For each k oo,

, tk -+ oo, we obtain

lim Xt < lim Et X = E3 +1-- dX , hence Xe < EB--kE9+1--dX.

This proves (2). Now let Xt be a uniformly integrable block martingale. Then (a) follows from part (2) applied to Xt and -Xt. Finally, to prove (b) assume that EtX = E t -- kEt +1-- dX for each k. Then E9--kXt

=

(b) (a),

E9--kEE --kEt+l--dX = Es--kEt+l--dX.

Let u be such that ui = ti for k+1 < i < d. Then as ul -+ oo, , uk --+ cc, we have Xu -+ Et +1 "" dX. Since E8 kXu =X(81, ,td), the proof is complete.

(9.4.3) Theorem. Let (Xt) be a uniformly integrable block submartingale, and write X = slim Xt. Then for all s, we have X. < 13,113 ... Es X . 9

Let (Xt) be a uniformly integrable block martingale. Then

X8 = E3E8 ...E.X. for all s.

Proof. Let (Xt) be a uniformly integrable block submartingale. We claim that for all k, 1 < k < d, (9.4.3a)

X. < ESX < E8 Ee ... E. E(k+1) -- dX.

9.4. Multiparameter martingales

399

Only the second inequality has to be proved. The proof is by induction on k.

Case k = 1 follows from (9.4.2). Now assume (9.4.3a) holds for some value of k, and consider the next value k + 1. Again by (9.4.2),

X8 < E8X < Taking the limit in L1 as Si -+ oo,

Es--(k+1)Ee+2--dX

, sk -+ 00,

E8k+1)--dX < Eek+1)Eek+2)--dX.

Now substitute this into (9.4.3a) to complete the induction step. This completes the proof in the case of a block submartingale. The case of a block martingale follows from this.

(9.4.4) Theorem. (i) Let (Xt) be a block submartingale that is bounded in the Orlicz space Llogd-1 L. Then we have upper demiconvergence: lim sup Xt = slim X. (ii) If X E L logd-1 L then Et Et Ed X converges a.e. to a finite limit. (iii) Let (Xt) be a block martingale bounded in L logd-1 L. Then Xt converges a.e. and in L logd-1 L.

Proof. (i) First assume that (Xt) is a positive block submartingale bounded in Llogd-1 L. Apply Theorem (9.1.3(ii)) and Lemma (9.4.1) with: L(i) = L log'-1 L for 1 < i < d; T(i, t) = Et. Note that as t1 -> oo, the a-algebras .Ft' increase to VtFl = Vt t, so Et'X converges to X as t1 --> oo. Hence T(1, oo) X = X a.e. Similarly T(i, oo) X = X a.e. for all i. Therefore, by (9.4.3) and (9.1.3(ii)), we have lim sup Xt < lim sup Et Et .

. . EdX

< T(1, oo) T(2, oo)

. T(d, oo) X = X.

On the other hand, we always have X = s lim Xt < lim sup Xt. Hence the statement follows for positive block submartingales; and hence also for block submartingales bounded from below by a constant. Now let Xt be an arbitrary L logd-1 L-bounded block submartingale. Then for any constant a, the process (Xt V a) is also a block submartingale,

so by the above, we have lim sup(Xt V a) = X V a. By Fatou's lemma, lim sup Xt > -oo a.e. Thus letting a -> -oo, we obtain lim sup Xt = X. (ii) Apply Theorem (9.1.3(iii)) and Lemma (9.4.1) with L(i) = L logx-1 L

for1,n for any fixed m, and the maximal inequality is preserved as m - -oo. Fix A E F1 with µ(A) < oo. In finite measure spaces, the localization theorem (1.4.2) states that X. = E-F' [X] for simple stopping times a. The conditional expectation is a contraction on L1(E) [see the remarks following (5.1.15)]; it follows that E [IIXII] 5 E [IIXII] Hence the maximal inequality (5.2.36) applied with the measure it restricted to A gives A

l1AsuIN

Now let A 10; it follows that Aµ {1A nSUP IIEY° [XI 11 >

A} < E [IIXII].

EIN

(b) From (8.2.5) we now obtain sup IIE--n[X]II > 2A} < E [1{IIXII?a}IIXII] ,t1A nEIN Now apply (9.2.1). (c) Xn converges on A by the convergence theorem (5.3.20) applied on the set A. Convergence on Sl follows on letting A T Q. (d) Write X = X, +X2, with X1 E L1 and IIX2IIoo < e. Since conditional

expectations are contractions in Lam, we may disregard X2. So we may assume simply that X E L1. But now the maximal inequality (a) allows the proof of convergence in the same way as in probability space (5.3.36).

Multiparameter processes

404

(9.4.10) (Multiparameter Rota theorem.) A positive operator is called bistochastic if it preserves the L1 and L,,. norms of positive functions. Let Ti be bistochastic operators, and let Un = Ti . T,,,,2 Tl . This is sometimes called the "alternating procedure." The one-parameter Rota theorem is: if f E R1, then Un f converges a.e. The operator admits a representation Un.f = E9

[E-F" [1]] ,

where 9 is a fixed v-algebra and .F is a decreasing sequence of v-algebras (Dellacherie & Meyer [1982], p. 56). In order to apply Theorem (9.1.3)

to obtain a multiparameter version of the theorem, f E Rk must imply sup Un f E Rk_1. This follows from the representation of U7, above, because (by Jensen's inequality) the conditional expectation E9 respects the classes

Rk. So the multiparameter Rota theorem is:

(9.4.10(i)). For i = 1,

,

d and n E IN, let Tn be a bistochastic operator.

Set

Un = T1 ...Tn

If f E Rd, then lim U31

U .d

)* ... (Ti)

f exists a.e. as the indices si converge to

infinity independently.

(9.4.11) (Pure LP alternating procedure.) Let T be a linear operator on LP, 1 < p < oo. Define a (nonlinear) operator M(T) : L4 - Lp by [(Tf)P-1])1/(P-1)

M(T)f = (T*

If (Tn)nEIN is a sequence of positive linear contraction operators on LP, then for all f E LP , the sequence M(T. Tl) f converges a.e. and sup. M(T. Tl) f E L4 (Akcoglu & Sucheston [1988]). If p = 2, then M(T) = T*T is linear. In that case, (9.1.3) implies an easy multiparameter version of (9.4.10(i)): Let (T,(, ))11EIN be sequences of positive contractions in L2, for i = 1,

If f EL2,then M(Tel) ... T21)T(1))M(T(2) ... T22)T12)) ... M(T(8dd) ...T(d)Tld)) f 2

converges a.e. as the indices s1, , sd converge to infinity independently. In particular, we obtain a multiparameter version of Stein's [1961] theorem, which corresponds to the case when the T,(,0 = T(i) are independent of n and self-adjoint. Then for each f E L2, (T(1))281 (T(2) J1\ 282

converges a.e. as the indices sl,

... (T(d)

284

)

f

, sd converge to infinity independently.

9.4. Multiparameter martingales

405

(9.4.12) (Amart approach to two parameters.) Let J = IN x IN. For t E J we write t = (t1i t2) and .':'t = V F(ti,u)u

Amapr:Sl --> J =IN xIN isa1-stoppingtime iff{z=t} EF' for all t E J. The set of simple 1-stopping times is denoted El. A 1-amart is a process (Xt) such that the net (E [XT])1-EEI converges. Assume that the stochastic basis satisfies the conditional independence condition (F4). Let (Xt) be a block martingale; equivalently, let (Xt) be a martingale and a 1-martingale (9.4.6). Then (Xt) is a 1-amart. Since (.F1) is totally ordered, this implies that Xt converges a.s. ((4.2.11) and (4.2.5)). Convergence of Xt also follows from (9.4.4) above, which is a more general argument. But the amart approach was particularly important in the continuous parameter case (J = IR x IR), where it was applied to the problem of regularity of trajectories of martingales bounded in L log L under condition (F4) (Millet & Sucheston [1981a]). This method proved regularity in the first, second, and fourth quadrants, after Bakry [1979] proved regularity in the first and third quadrants by the method of stochastic integration; see also Meyer [1981]. Again the amart point of view gave the good notions of optional and predictable projections for two-parameter processes; see Bakry [1981]. (9.4.13) (Multiparameter Krengel theorem.) Krengel's stochastic ergodic theorem asserts that if T is a positive linear contraction on L1 and f E L+1, then (1/n) E O Ti f converges stochastically. Since n-1 lim inf

1 >2 Ti f = 0, i-o

there is in fact lower demiconvergence to 0 (Krengel [1985], p. 143). Theorem (9.1.3(i)) now gives a multiparameter version of this result (Millet & Sucheston [1989]).

(9.4.14) (Additive amarts.) Multiparameter processes may be studied in another way. A few details are given here. Let I be a directed set with least element 0, and locally finite in the sense

that all intervals [0, t] = { s E I : 0 < s < t } are finite. A subset Sc I is a (lower) layer of I if from s < t and t E S it follows that s E S. We will write G(I) = L for the set of all layers. Then L is a directed set when ordered by inclusion. So we may study processes with index set L, such as martingales or amarts. If (.Ft)tEI is a stochastic basis indexed by I, there is an associated stochastic basis (?Is)sE.c indexed by the layers, defined by

is = V .F8. 8ES

A process Fs indexed by G is called an additive process if FSUT + FsnT = Fs + FT

a.e.

Multiparameter processes

406

for all S, T E L. Certainly any process (FS) of the form

FS = > Y,, sES

is additive. An additive process that is also an amart is called an additive amart. An additive process that is also a martingale is called an additive martingale. If a process (Xt)tEI has the form

Xt=>YB s