Probabilities and Potential

192 Pages·1979·8.1 MB·English

by  Claude Dellacherie

NORTH-HOLLAND
MATHEMATICS STUDIES 29
Probabilities and Potential
CLAUDE DELLACHERIE
Institut de Mathematique
Universite Louis-Pasteur, Strasbourg
PAUL-ANDRE MEYER
Directeur de recherches
Centre National de la Recherche Scientifique
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I
HERMANN. PUBLISHERS IN ART AND SCIENCE
1978
NORTH-HOLLAND PUBLISHING COMPANY - AMSTERDAM - NEW YORK - OXFORD
© Hermann, Paris 1978
All rights reserved. No part ofthis publication may be reproduced, stored in a retrieval system,
or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording
or otherwise, without the prior permission of the copyright owner.
Hermann ISBN: 2 7056 5857 2
North-Holland ISBN: 0 72040701 x
Translation of: Probabilites et potentiel © 1975, Hermann, 293 rue Lecourbe, 75015 Paris_ France
PUBLISHERS
HERMANN, PARIS
NORTH-HOLLAND PUBLISHING COMPANY
AMSTERDAM • NEW YORK • OXFORD
SOLE DISTRIBUTORS FOR THE U.S.A. AND CANADA:
ELSEVIER NORTH-HOLLAND, INC.
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Library of Congress Cataloging in Publication Data :
Dellacherie, Claude.
Probabilities and potential.
(North-Holland mathematics studies; 29)
Translation of Probabilites et potentiel.
Ed. of 1966 by P.-A. Meyer
Bibliography: p
Includes indexes.
1. Probabilities. 2. Measure theory. 3. Poten
tial, Theory of. I. Meyer, Paul Andre, joint author.
II. Meyer, Paul Andre. Probabilites et potentiel.
III. Title.
QA273.D3713 519.2 77-26865
ISBN 0-7204-0701-X
PRINTED IN FRANCE
Contents
CHAPTER O. NOTATION
1
O-v\PTER I. f''EASURABLE SPACES
7
a-fields and random variables 7
Definition of a-fields (no.l). Random variables (nos. 2 to 4). a-fields
generated by subsets, etc... (nos. 5 to 7). Product a-fields (no. 8). Atoms,
separable a-fields (nos. 9 to 12).
Real-valued random variables 11
First properties (nos. 13 to 18). The monotone class theorem (nos. 19 to 24).
CW\PTER II. PROPABILITY lAWS AND mTHEMl\TlCAL EXPECTATIONS
17
Summary of integration theory 17
Probability laws (nos. 1 to 4). Expectations, Lebesgue's theorem, etc...
(nos. 5 to 9). Convergence of random variables (no. 10). Image laws (nos. 11
to 12). Integration of laws, Fubini's Theorem (nos. 13 to 16).
Supplement on integration
21
Oniform integrability (nos. 17 to 22). Vitali-Hahn-Saks Theorem (no. 23).
Weak compactness, Dunford-Pettis Compactness Theorem (nos. 24 to 26). Rapid
filters (nos. 27 to 29).
Completion, independence, conditioning 30
Internally negligible sets (nos. 30 to 32). Independence (nos. 33 to 35).
Conditional expectations (nos. 36 to 40). List of their properties (nos 41 to
42). Conditional independence (nos. 43 to 45).
CHAPTER III. CCl''PLEmfTS TO rtEASURE THEORY 39
Analytic sets 39
Pavings (no.l) Compact pavings (nos. 2 to 6). Analytic sets and closure pro
perties (nos. 7 to 13). The separation theorem (no. 14). Souslin measurable
spaces, etc... (nos. 15 to 17). Direct images (nos. 18 to 20). The Souslin-
Lusin theorem (nos. 21 to 23). Blackwell spaces (nos. 24 to 26).
Capacities 5
Choquet capacities (no. 27). Choquetls theorem (nos. 28 to 29). Construction
of capacities (nos. 30 to 32). Applications: measurability of analytic sets
(no. 33), Caractheodory's theorem (no. 34) and Daniell IS theorem (no. 35);
measures on compact spaces (no. 36) and Lusin spaces (nos. 37 to 38). Left
continuous (nos. 39 to 40) and right-continuous (nos. 41 to 42) capacities.
Another proof of the separation theorem (no. 43). Measurability of debuts and
cross section theorem (nos. 44 to 45).
Bounded Radon measures
6!
Radon measures (nos. 46 to 47). Filtering famili~s of l.s.c. or u.s.c.
functions (nos. 48 to 50). Inverse limits: Kolmogorov's theorem ~os. 51
to 52) and Prokhorov's theorem (no. 53). Strict convergence (nos. 54
to 58). Prokhorov's compactness criterion (no. 59). The space of probability
laws (nos. 60 to 62). Lindelof spaces (nos. 63 to 66). Non-metrizable Souslin
and Lusin spaces (nos. 67 to 69). Desintegration of measures (nos. 70 to 74).
(N0S 75 to 86 appear in the appendix, see below).
CHAPTER IV STOCHASTIC PROCESSES 8~
I
General properties of processes 8~
Processes (no. 1). Philosophy (nos. 2 to 5). Standard modifications and
indistinguishable processes (nos. 6 to 8). Time laws canonical processes
(nos. 9 to 10). Filtrations, adapted processes and philosophy (nos. 11 to 13).
Progressivity (nos. 14 to 15).
Regularity of paths 85
Notation- (no. 16). Processes on a countable dense set (nos. 17 to 19).
Upcrossings and downcrossings, applications (nos. 20 to 23). Separable processes
(nos. 24 to 30). Random closed sets (nos. 31 to 32). Progressivity of certain
processes (nos. 33 to 34). Almost-equivalence (no. 35). Essential topology
(nos. 36 to 39). Pseudo-paths (nos. 40 to 46).
Optional and predictable times
11
Definitions concerning filtrations (nos. 47 to 48). Stopping times (no. 49).
Debuts (nos. 50 to 51). a-fields associated to stopping times (nos. 52 to 54).
Properties (nos. 55 to 59). Stochastic intervals (no. 60). Optinal and predie
table a-fields (nos. 61 to 63). Properties (nos. 64 to 68). Predictable times
(nos. 69 to 74). Sequences foretelling a predictable time (nos. 75 to 78).
Totally inaccessible stopping times; classification (nos. 79 to 81). Quasi
left-continuous filtrations (nos. 82 to 83). Cross section theorems (nos 84
to 87). Sets with countable sections (no. 88).
Additional numbers on sets with countable sections (nos
Examples and supplements 143
Optional or predictable processes defined by limits (nos. 89 to 93). Canonical
spaces (nos. 94 to 96), their predictable and optional a-fields (nos. 97 to 98)
and Galmarino's test (nos. 99 to 102). Decomposition of a stopping time (no.
103). Filtrations generated by a single stopping time (nos. 104 to 108).
APPENDIX TO CHAPTER III 156
Souslin schemes (nos. 75 to 77). Representation of Souslin and Lusin spaces as
continuous images of Polish spaces (nos. 78 to 80). Isomorphisms:between Lusin
spaces (no. 80). Cross section theorem (nos. 81 to 82). The second separation
theorem (83).
APPENDIX TO CHAPTER IV 163
Sets with un-countable sections (nos. 109 to 113). Derivatives of random sets
(nos. 114 to 116). Sets with countable sections (nos. 117 to 118).
CO't'ENTS 169
INDEX OF TERJ~INOLOGY 175
H'lDEX OF ['{)TA11or~ 181
BlEUOGRAPHY 185
Preface
The titles of most books are meant to provide some information about their
contents. So it is only fair to warn the reader that this volume contains little
enough probability and no potential theory whatsoever (1). Most of the probability
should appear in a subsequent volume (martingale theory), potential theory still
later (resolvents and semi-groups). As for the "and" between these two words, it is
pushed so far into the future that we scarcely dare think about it. The true
contents of this volume are some brief recollections of measure theory and the
vocabulary of probability, and two long chapters, the first one on analytic sets
and capacities, and the second one on the foundations of stochastic processes.
Why then this title? In the year 1966, the second author had already publi
shed a volume called Probability and Potentials, containing eleven chapters which
covered a much wider domain. It only lacked the "and", that is, the connection
between potential theory and the theory of Markov processes. This was meant for a
second volume, whose partial outline appeared in 1967 as a set of lecture notes on
Markov processes. And now, instead of completing this second part to crown the
edifice, we return to the very foundations of it. This may look absurd, but there
are several reasons for it. First of all, the need for a reference book on Markov
precesses and potential theory which was felt in those times was relieved by the
publication, in 1968, of the treatise of Blumenthal and Getoor. Next, our whole
theory has been since 1966 in a process of very rapid evolution. To take a few
examples, in probability theory the first edition of this book contained the defi
nition of a well-measurable process, but that of a predictable process, which is
now so basic for stochastic integration, was only implicit there. On the potential
side, a modest notion called "pseudo-reduite" (p. 247 of the French edition) was
(1) So we are following in the footsteps of our Master N. Bourbaki, whose stpuctures
fondamentales de 1'analyse contain no analysis at all.
introduced with the somewhat despising comment "we aren't sure that the following
theorem can be of any use". From the work of Mokobodzki on resolvents, we have
learnt since that (pseudo)reduites are the key to the deeper results of potential
theory. Finally, concerning the "and" part, the announcement at the beginning of
the section on Ray resolvents was "the following results will not be used in later
chapters, and can be omitted", while they would now be considered as fundamental.
Examples of this kind could be multiplied.
The conditions of our work have also changed considerably since 1966. At that
time, whereas potential theory and the theory of Markov processes were respectable
areas of mathematics, people interested in the relation between them would scarcely
outnumber half a score in the world. This is no longer the case (may be some credit
for it can be ascribed to the first edition of this book which, for all its imper
fections, has contributed to popularizing a number of ideas). There are just two
names on the cover, but this shouldn1t hide the fact that the new points of view
presented here, or to be presented in later volumes, came into being through
innumerable exchanges. The reader will gain some idea of it by perusing the volumes
of the Strasbourg probability seminar, published every year since 1967 - and this
is but the tip of the iceberg.
Thus the rapid evolution of the whole theory has discouraged us from building
on the old foundations, and the support of an active mathematical environment has
been an incentive to undertake again the full work from the start. Our publisher
also has been full of understanding in his acceptance of a publication "by instal
ments", more informal than usual.
From the history of our theory we have also learnt some lessons, which we have
tried to put to use in this new edition. In particular, we have tried to free our
selves from the attitude of many textbooks, which deal with mathematical truths as
with eternal objects offered to our contemplation, from a world of pure Ideas where
inflation is something unheard of. Truths are truths, but their value doesn't come
from being printed on fine paper. Many immutable truths of 1966 have lost all
interest and are now dead, while small remarks of 1966 have grown up and now shed
light on large parts of our field.
So we have tried to put ~s much life as we could into the work, making
digressions, adding comments and leaving some room for technical tricks and
"useless" remarks. We must confess that the material may be considered arid, and
that boredom has overcome us at times (at which places, the reader will probably
know by his own weariness), but not too often.
We have preserved the organization of the first edition : within each chapter,
all statements (whether theorems, definition or remarks) to which it may be useful
to refer are sequentially numbered. So Theorem II. 31 (II denotes the chapter) may
be followed by remarks II. 32 a) and b), and Definition II. 33. This is convenient
for the reader (so we believe), but the cost to the authors of modifying a chapter
that is almost completed becomes enormous. So we beg from our readers some indul
gence for irregularities: "bis" and missing numbers, or maybe trivial remarks
glorified with a number to prevent a gap in the numbering. Indexes of notation and
terminology at the end of the volume are organized according to this system, not to
page numbers. The bibliography is classified by alphabetical order of authors, but
in the list of each author's publications (numbered [1J, [2J ...) the order is
purely random, as fits a probability book.
We should have dedicated the book to our wives, for keeping the children quiet
while Daddy was working (or pretending to), but we got from Frank Knight the (secret)
information that 1976 was the year of Doob's 65 th birthday (1). Now Doob's ideas
inspired a great deal of the work in our field and in particular pervade the whole
of our chapter IV. So it was only justice to write here:
DEDICATED TO J.L. DOOB ON HIS 65 TH BIRTHDAY
(1) Our hearty thanks go to Professor T.G. Kurtz, who helped us to prepare the
final manuscript of the English edition. His comments, on mathematics and language,
led to the elimination of many errors and obscurities.
CHAPTER 0
Notation
Notation from set theory 1
C
The complement of A is denoted by [A or more often A . The notation A\B means
An BC; A~ B is the symmetric difference (A\B) U (B\A). The set of all x E Ewith
some property P is denoted by {x E E : P(x)} or, if there is no ambiguity, {x:P(x)}
or simply {Pl.
The restriction of a function f to a set A is denoted by fi . Similarly, if t
A
e :
is a family of subsets, ciA is the set of traces on Aof elements of
explicitly el {B () A, BEn.
=
A
Closure of sets of subsets 2
e
We sometimes use sentences of the following form: the family is closed
under (...), where the brackets contain set-theoretic operation symbols, sometimes
followed by the letters f, c, a, m, which abbreviate respectively: finite, countable,
arbitrary, monotone. Two examples will suffice to clarify their meaning: lie is
closed under (uf,na)" means that finite unions (*) of elements of 8 and arbitrary
intersections of elements of e still belong to E; lie is closed under (umc, cr' means
e
that monotone countable unions of elements of (i .e. unions of increasing sequences
ine) still belong toe and that complements of elements of ~still belong
to e. Sets of subsets or functions are generally denoted by capital script letters.
e eo
The closure of a family of subsets under (uc) (resp. (nc)) is denoted by
(resp.80) - this notation is classical to set theory. We write ((e)a)o = eao '
Lattice notation 3
Let f and 9 be two real-valued functions. We write f v g and fAg for sup(f,g)
and inf(f,g). The notation f+ and f- has its classical meaning: f+ f 0, f-
= v =
(-f) v O.
More generally, V,A denote least upper and greatest lower bounds: for example,
v
the a-field generated by the union of a family of a-fields J is denoted by i 1 .
i i
(*) Bourbaki includes under finite unions the "empty union" and similarly for inter
sections. We do not use this convention.
2-0 PROBABILITIES
4 Limits along Rand N
The notation s t t means s t, sst; s t tt means s t, s t ; sn t t,
-+ -+ <
sn ttt is used similarly for sequences (sn)' with the additional meaning that (sn)
is increasing. Obvious changes are required if + appears instead of t. The usual
notations lim ,lim inf for limits along N will be written simply as lim,
n
Y)-+<lO Y)-+<lO .
1im inf .
n
5 Integration theory
The word measure without further qualification always means IIpositive (1)
countably additive set function on an abstract measurable spacell We do not adhere to
•
the convention that all measures considered are a-finite: this would cost us too much
generality in potential theory but weill consider only countable sums of bounded
measures (no difficulties relating to Fubini IS theorem, for example, can arise with,
such measures).
The notation Av ~ = SUp(A,~),A A ~ = inf(A,~), ~+, ~- ,I~I = ~+ + ~- has its
classical meaning. II~II denotes the total mass <I~I ,1> of ~ (sometimes infinite). The
~ ff(x)~(dx)(2)
integral of a function f with respect to a measure is denoted by :
ff~ ~ withou~
often abridged into ; denotes a Radon-Nikodym density, IIdll However
•
when a measure ~ on R appears as the derivative of an increasing function F, we use
the standard notation with IIdll for Stieltjes integrals ff(x)dF(x), and in particular
ff(X)dX if F(x) = x. J
~ ff~ fAf~
If is a probability law, we often write [[f] for and [[f,A] for
(especially when A is a complicated event).
6 Function spaces
If E is a topological space, e(E), Cb(E), Cc(E), CO(E) denote the spaces of
real-valued functions which are respectively continuous, bounded and continuous,
continuous with compact support, continuous and tending to 0 at infinity (the latter
when E is locally compact). Adjoining a + to this notation (C+(E) and so on) enables
us to denote the corresponding cones of positive functions. As usual CC00(E) denotes
the space of infinitely differentiable functions with compact support (E then is as
n
sumed to be a manifeld, usually R ).
If (E,~) is a measurable space, the notation ~(e) (resp. b(e)) denotes the
space of e-measurable (resp. bounded e-measurable) real functions. Spaces of measures
are used only on a Hausdorff topological space E :17L~(E), 1n+(E) then are the cones of
bounded (resp. arbitrary) Radon measures on E and 17I (E), !7L(E) are the vector spaces
b
generated by these cones.
(1) Positive means ~O, according to European use.
(2) Also ~(f), <~,f>.