Table Of Content

MATHEMATICAL FOUNDATIONS OF THE CALCULUS OF PROBABILITY HOLDEN-DAY SERIES IN PROBABILITY AND STATISTICS E. L. Lehmann, Editor MATHEMATICAL FOUNDATIONS OF THE CALCULUS OF PROBABILITY By JACQUES ~EVEU Faculty of Sciences, University of Paris Translated by AMIEL FEINSTEIN Foreword by R. FORTET na HOLDEN-DAY, INC. San Francisco, London, Amsterdam 1965 --519. I t/. .NS1r '1"· :;_ c This book is translated from Bases Mathematiques du Ca/cul des Probabilites, 1964, Masson et Cie, Paris © Copyright 1965 by Holden-Day, Inc., 728 Montgomery Street, San Francisco, California. All rights reserved. No part of this book may be reproduced in any form without permission in writing from the publisher Library of Congress Catalog Card Number: 66-11140 Printed in the United States of America / FOREWORD In its present state, the calculus of probability and, in particular, the _J_heory of stochastic processes and vector-valued random variables, cannot be understood by one who does not have, to begin with, a thorough understanding of measure theory. If one is to prepare for participation in the future development of thec alculus of probability, it is not sufficient to know the fundamental concepts and results of measure theory; one must also be experienced in its techniques and able t~hem and extend them to new situations. Again, one often hears-and quite justifiably in a certain sense-that the calculus of probability is simply a paragraph of the theory of measure; but within measure theory, the calculus of probability stands out by the nature of the guestions which it seeks toa nswer- a nature which has its origins not in measure theory itself, but in the philosophical and practical content of the notion of probability. The advanced course in the calculus of probability is aimed at those students having the body of knowledge which in France is called "licence de Mathematiques"; this body of knowledge covers mathematics in general. It naturally encompasses the theory of measure and integration, but is necessarily limited to an introduction to the subject. It is thus necessary that this question be taken up again and developed in advanced studies; it is still more necessary that its exposition be oriented specifically toward applications to probability theory. Since 1959, Professor Neveu has been given the task of presenting this course at the Faculty of Sciences of Paris. It is not necessary to introduce him to the specialists in probability theory. In a short number of years he has gained their attention by brilliant work; but not all can know as well as I bow much our students and young researchers appreciate his lively v vi FOREWORD and clear method of teaching. The course which he has taught, enriched by this pedagogic experience, constitutes the subject matter of the present work. To be sure, there are already books, some of them more extensive, on the theory of measure and several of them are excellent. However, I have already stated why probabilists have need of a text written especially for their use; and for beginners, a text of limited size is preferable. In such a domain, Professor Neveu has naturally sought to write an expository book, not one of original work, in the sense that he does not pretend to introduce new concepts or to establish new theorems. The fact that he has, very usefully, enriched each chapter with Comple ments and Problems underlines the essentially pedagogic objective of his book, concerning which I can with pleasure point out two non-trivial merits: he avoids an overburdened notation, and, in a subject which is by its nature abstract, he does not hesitate to insert whenever necessary a paragraph which interprets, which states the reason for things, or which calls attention to an error to be avoided. The exposition nevertheless proceeds with profound originality. First, by its contents: To the classical elements of measure and integration, the author adds all the theorems for the construction of a probability by extension; from an algebra to a a-algebra, from a compact subclass to a semialgebra, from finite products of spaces to infinite products of spaces (theorems of Kolmogorov and Tulcea), etc. He treats the measurability, separability and the construction of random functions; conditional ex pectations, and martingales. He illustrates general results by applications to stopping times, ergodic theory, Markov processes, as well as other problems, all of these rarely included in treatises on measure theory, some of them because of their recent development, others because, while they are of major importance in probability theory, they are perhaps of less interest in general measure theory. The originality appears equally in the presentation; I particularly appreciate the simple but systematic way in which Professor Neveu has set forth from the first the algebraic structures of families of events which intervene (Boolean algebra and a-algebras, etc.), while avoiding the pre mature introduction of topological concepts, whose significance is thereby even better understood. Throughout, he has succeeded in establishing the most concise and elegant proofs, so that in a small number of pages he is able to be remark ably complete; for example he treats, at least briefly, LP spaces and even, FOREWORD vii by a judicious use of the Complements and Problems, decision theory and sufficient statistics. As a text for study by advanced students, as a reference work for researchers, I can without risk predict for this book long life and great success. R. Fortet August, 1965 Geneva TRANSLATOR'S PREFACE In comparison with the French original, this translation has benefited by the addition of a section (IV.7) on sequences of independent random variables, as well as by certain additions to the Complements and Problems. Also, the proofs of a few results have been modified. For these improve ments, and in particular for the full measure of assistance which I have received from Professor Neveu at every stage of the translation, it is a pleasure to record here my deep gratitude. A. Feinstein DEFINITIONS AND NOTATION Definitions of the terms partially ordered set (or system), totally ordered set, lattice, complete lattice, generalized sequence, vector space, Banach space, and linear functional (among others used in this book) may be found in Chapters I and II of the treatise Linear Operators, Part I, by N. Dunford and J. T. Schwartz. A real vector lattice is a set which is both a lattice (under some partial ordering) and a real vector space, and such that x ~ y implies ex ~ cy for every real c > 0, and also z + x ~ z + y for every z. A linear func tional f on a vector lattice is said to be positive if x ~ 0 implies f(x) ~ 0. A partially ordered set Eis said to be inductive if it satisfies the hypothesis of Zorn's lemma (Dunford and Schwartz, p. 6), i.e., if every totally ordered subset of E has an upper bound in E. A pre-Hilbert space is a space satisfying all the axioms of a Hilbert space except the axiom of completeness. viii TRANSLATOR'S PREFACE ix The symbol :::> indicates logical implication; I denotes the end of a proof; {x: · · ·} denotes the set of all objects x which satisfy the conditions · · · ; * marks difficult sections or problems, for whose understanding or = solution concepts not discussed in the text may be needed; finally, has occasionally been used for "equality by definition." AUTHOR'S PREFACE The object of the theory of probability is the mathematical analysis of the notion of chance. As a mathematical discipline, it can only develop in a rigorous manner if it is based upon a system of precise definitions and axioms. Historically, the formulation of such a mathematical basis and the mathematical elaboration of the theory goes back to the 1930's. In fact, it was only at this period that the theory of measure and of integration on general spaces was sufficiently developed to furnish the theory of probability with its fundamental definitions, as well as its most powerful tool for development. Since then, numerous probabilistic investigations, undertaken in the theoretical as well as practical domain, in particular those making use of functional spaces, have only served to confirm the close relations established between probability theory and measure theory. These relations are, incidentally, so close that certain authors have been loath to see in proba bility theory more than an extension (but how important a one!) of measure theory. In any case, it is impossible at the present time to undertake a pro found study of probability theory and mathematical statistics without continually making use of measure theory, unless one limits oneself to a study of very elementary probabilistic models and, in particular, cuts one self off from the consideration of random functions. Attempts have been made, it is true, to treat convergence problems of probability theory within the restricted framework of the study of distribution functions; but this procedure only gives a false simplification of the question and further conceals the intuitive basis of these problems. The book reproduces the essentials of a course for the first year of the third cycle (which corresponds roughly to the first or second year of x