Table Of ContentProbability Distributions on Banach Spaces
Mathematics and Its Applications (Soviet Series)
Managing Editor:
M. HAZEWINKEL
Centre jor Mathematics and Computer Science, Amsterdam, The Netherlands
Editorial Board:
A. A. KIRILLOV, MGU, Moscow, U.S.S.R.
Yu. 1. MANIN, Steklov Institute of Mathematics, Moscow, U.S.S.R.
N. N. MOISEEV, Computing Centre, Academy of Sciences, Moscow, U.S.S.R.
S. P. NOVIKOV, Landau Institute of Theoretical Physics, Moscow, U.S.S.R.
M. C. POLYVANOV, Steklov Institute oj Mathematics, Moscow, U.S.S.R.
Yu. A. ROZANOV, Steklov Institute oj Mathematics, Moscow, U.S.S.R.
Probability Distnbutions
on Banach Spaces
by
N. N. Vakhania
V. 1. Tarieladze
and
S. A. Chobanyan
Academy of Sciences of Georgian S.S.R.,
Institute of Computer Mathematics,
Tbilisi State University, U.S.S.R.
Translated by Wojbor A. Woyczynski
D. Reidel Publishing Company
A MEMBER OF THE KLUWER ACADEMIC PUBLISHERS GROUP
Dordrecht / Boston / Lancaster / Tokyo
Library of Congress Cataloging in PubHeation Data
Vakhania, N. N. (Nikolai Nikolaevich)
Probability distributions on Banach spaces.
Mathematics and its applications (D. Reidel Publishing Company). Soviet series
Translation of: Veroiatnostnye raspredeleniia v banakhovykh prostranstvakh.
Bibliography: p.
Includes index.
1. Probabilities. 2. Banach spaces. I. Tarie1adze, V. I. (Vazha Izemovich)
II. Chobanyan, S. A. (Sergei Akopovich) III. Title. IV. Series.
QA273.V3313 1987 515.7'32 87-4931
ISBN 90-277-2496-2
ISBN-13:978-94-010-8222-8 e-ISBN_13:978-94-009-3873-1
DOl: 10.1007/978-94-009-3873-1
Published by D. Reidel Publishing Company,
P.O. Box 17, 3300 AA Dordrecht, Holland.
Sold and distributed in the U.S.A. and Canada
by Kluwer Academic Publishers,
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In all other countries, sold and distributed
by Kluwer Academic Publishers Group,
P.o. Box 322,3300 AH Dordrecht, Holland.
All Rights Reserved
© 1987 by D. Reidel Publishing Company, Dordrecht, Holland
No part of the material protected by this copyright notice may be reproduced or
utiIized in any form or by any means, electronic or mechanical
including photocopying, recording or by any information storage and
retrieval system, without written permission from the copyright owner
SERIES EDITOR'S PREFACE
Approach your problems from the right end It isn't that they can't see the solution. It is
and begin with the answers. Then one day, that they can't see the problem.
perhaps you will find the final question.
G.K. Chesterton. The Scandal of Father
'The Hermit Clad in Crane Feathers' in R Brown 'The point of a Pin'.
van Gulik's The Chinese Maze Murders.
Growing specialization and diversification have brought a host of monographs and
textbooks on increasingly specialized topics. However, the "tree" of knowledge of
mathematics and related fields does not grow only by putting forth new branches. It
also happens, quite often in fact, that branches which were thought to be completely
disparate are suddenly seen to be related.
Further, the kind and level of sophistication of mathematics applied in various
sciences has changed drastically in recent years: measure theory is used (non
trivially) in regional and theoretical economics; algebraic geometry interacts with
physics; the Minkowsky lemma, coding theory and the structure of water meet one
another in packing and covering theory; quantum fields, crystal defects and
mathematical programming profit from homotopy theory; Lie algebras are relevant
to filtering; and prediction and electrical engineering can use Stein spaces. And in
addition to this there are such new emerging subdisciplines as "experimental
mathematics", "CFD", "completely integrable systems", "chaos, synergetics and
large-scale order", which are almost impossible to fit into the existing classification
schemes. They draw upon widely different sections of mathematics. This pro
gramme, Mathematics and Its Applications, is devoted to new emerging
(sub)disciplines and to such (new) interrelations as exempla gratia:
- a central concept which plays an important role in several different mathematical
and/ or scientific specialized areas;
- new applications of the results and ideas from one area of scientific endeavour
into another;
- influences which the results, problems and concepts of one field of enquiry have
and have had on the development of another.
The Mathematics and Its Applications programme tries to make available a careful
selection of books which fit the philosophy outlined above. With such books, which
are stimulating rather than definitive, intriguing rather than encyclopaedic, we hope
to contribute something towards better communication among the practitioners in
diversified fields.
v
vi SERIES EDITOR'S PREFACE
Because of the wealth of scholarly research being undertaken in the Soviet
Union, Eastern Europe, and Japan, it was decided to devote special attention to the
work emanating from these particular regions. Thus it was decided to start three
regional series under the umbrella of the main MIA programme.
Probability has always had an enormous range of applications in the various sci
ences. and lately it has also began to interact more fundamentally with such older
branches of mathematics as analysis. geometry and algebra.
Just as there arose in these branches the need for infinite-dimensional generaliza
tions so is true in probability. Except that here it happened earlier. Indeed. the fun
damental stochastic process. Brownian motion or the Wiener process. which is at
the basis of practically all stochastic analysis and geometry and the applications
thereof in physics. chemistry, biology and the geological sciences, is after all a meas
ure on an infinite-dimensional space of paths. Thus, infinite-dimensional spaces
come up very quickly - as soon as one becomes interested in the probability of
paths and surfaces and ...• instead of just probabilities over sets of points. Also.
applications of these ideas range from physics (e.g. the Polyakov measure on spaces
of Riemann surfaces in (super)string theory) to pattern recognition and mathemati
cal morphology.
As the translator notes in his preface, the subject has reached a certain maturity
and the present volume is the first book appearing at this stage. As he also notes,
the book does not. by design. deal with the large and more specialized topic of cen
trallimit theorems and related matters. The authors write that they consider that to
be material for a companion volume. And in fact such a "companion volume" is
being planned for this series (by Paulauskas and Rackauscas).
In many of my students and colleagues I have often noted a certain reserve
when their considerations threatened to start involving infinite-dimensional proba
bility spaces and path integrals and such: much like a swimmer at the edge of a lake
who feels sure the water will be a great deal colder than everybody assures him it
will be. This book can do much to remove that reserve.
The unreasonable effectiveness of mathemat As long as algebra and geometry proceeded
ics in science ... along separate paths, their advance was slow
and their applications limited.
Eugene Wigner But when these sciences joined company
they drew from each other fresh vitality and
Well, if you know of a better 'ole, go to it. thenceforward marched on at a rapid pace
towards perfection.
Bruce Bairnsfather
Joseph Louis Lagrange.
What is now proved was once only ima
gined.
William Blake
Trieste. 22 February 1987 Michie1 Hazewinkel
TRANSLATOR'S PREFACE
This book is the latest in a series of attempts to
present a systematic foundation of the theory of
probability measures on infinite-dimensional spaces, and
perhaps the first written at the stage when the theory has
reached relative maturity. The coming of age of the theory
was preceded and inspired by, among others, a number of
pioneering monographs including those by V. Grenander
(1963), K. Parthasarathy (1967), P. Billingsley (1968),
L. Schwartz (1973) and by Professor Vakhania (1971)
himself.
The present volume is quite comprehensive as far as
the foundations are concerned but, by design, it does not
cover more specialized topics such as the central limit
problem and the geometric aspects of the theory. An
interested reader is advised to consider expositions by
A. Araujo and E. Gine (1980) and by the translator (1978)
for their coverage.
The book can certainly serve as an excellent reference
monograph containing a great deal of material which, until
now, was scattered in a plethora of journal articles,
including those published only in Russian and largely
inaccessible to the Western audience. Also, a large number
of diversified exercises contained in the book indicates
that it has a considerable potential as a textbook for
someone trying to get involved in research in infini te
dimensional probability theory.
The authors themselves contributed in a significant
way to the developmen t of the theory and continue to be
active researchers in the area covered by the book. A
seminar formed by them in Tbilisi, Georgia, became one of
the internationally renowned centers of research on
probability theory in Banach spaces.
The camera-ready copy of the translation was prepared,
with great attention to detail, by Joyce Pluth, and Alicja
Szulga magnanimously helped with proofreading and the
subject index.
W.A.W.
Cleveland, Ohio, September 1986
vii
FOREWORll
In recent times the language and concepts of
probability theory found applications in a great variety of
disciplines, both inside and outside mathematics. To a
large extent this phenomenon can be explained by the fact
that in the last few decades many classical results of
probability theory have been extended to more general
structures, including the important case of Banach spaces.
The present book is devoted to general problems of the
theory of probability distributions on Banach spaces. We
believe that more special topics such as Gaussian, stable .
and infinitely divisible measures, the central limit
theorem, the laws of large numbers and a study of
connections between probabilistic properties and the
geometry of Banach spaces should be the subject of a
separate book. In this book we do encounter some of these
problems but they are playing only a secondary role and are
not an independent object of study.
Probability distribution is just another term for a
probability measure i.e. a finite, non-negative, countab1y
additive function on a a-algebra of subsets which is equal
to one on the whole set (also terms "probability",
"t'IJ~oj.r~.f.fu\.t;!..cm~', "i\iN~J."lYi\tC.l\Tl.J. \1'.[ -pro-lJa-di.ii1!1e~'- are
sometimes used). In the title we favored the term
"distribution" over the term "measure", emphasizing the
fact that probability measures are of our interest primarily
as distributions of random elements.
The exposition of material is organized, so to speak,
on three levels: 1) basic text containing a systematic,
sequential and complete presentation of the main body of
material; 2) supplementary comments which close each of
six chapters of the book and which contain oibliographica1
references and information concerning generalizations and
extensions of the subject matter discussed in the
corresponding chapter. The material of supplementary
comments does not pretend to be complete, final or
connected; in the basic text this material is not being
used. Understandably, the selection of supplementary
material was limited not only by the scope of our book but
ix
x FOREWORD
also by our own tastes and by the literature available to
us; 3) exercises (there are some three hundred of them in
the book) which end each of 27 sections of the basic text
and which provide not only the usual training material but
also enlighten and deepen the basic material. Whenever
deemed necessary, exercises are augmented by hints.
To comprehend the book, besides the obvious necessary
desire to understand the subject, it is sufficient for the
reader to have some background in the elements of
functional analysis and in the theory of integration of
real-valued functions on abstract measure spaces.
Familiarity with the classical probability theory is
desirable but, formally, it is not necessary.
The book uses the following system of references:
Proposition (Theorem, Lemma) 1.2.3 refers to Proposition 3
of Section 2 of Chapter I. If the statement we are
referring to comes from the same chapter (section) then the
first number (two first numbers) are omitted.
The maj ority of resul ts presented in this book (and,
in particular, results obtained by the authors) were
discussed in one form or another at the seminar on
probability distributions on functional spaces held in the
N. I. Muschelishvili Institute of Computational Mathematics
of the Academy of Sciences of the Georgian SSR. The
au thors would like to expres s their gra ti tude to all the
participan ts of the seminar and, in particular, to A. G.
Shangua, who carefully went through the entire manuscript
and helped us to avoid a number of inaccuracies.
Authors
TABLE OF CONTENTS
TRANSLATOR'S PREFACE................................ vii
·FOREWORD. • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • • l.X
ABBREVIATIONS AND SyMBOLS........................... xvii
INTRODUCTION •••••••••••••••••• xxiii
II • • • • • • • • • • • • • • • • • • • • •
Chapter I. MEASURABILITY AND MEASURES.............. 1
1. a-algebras and measurable mappings in metric
.space........................................ 1
1. Measurable spaces and mappings, Borel
a-algebra (1). 2. a-algebras generated by
families of mappings (3). 3. a-algebras in
product spaces (8). 4. The structure of
measurable mappings (10). Exercises (14).
2. a-algebras in Banach spaces.................. 15
1. Measurable vector spaces (15).
2. Relations between various a-algebras (16).
3. Auxiliary results and counter-examples
(20). Exercises (24).
3. Probability measures on topological spaces... 25
1. Preliminaries from the general measure
theory (25). 2. Regular, T-smooth and Radon
measures (27). 3. The support of a measure
(30). 4. Existence of T-smooth and Radon
extensions (33). 5. Extensions of Baire
measures. Prokhorov's theorem (38).
6. Embedding of the space of measures into
the space of linear functionals (40).
7. Weak convergence of probability measures
(41). 8. The weak topology. Prokhorov's
criterion (47). 9. Criteria of weak relative
compactness of families of probability
measures on a Banach space (51). Exercises
(54).
