Patrick Billingsley - Probability and Measure

Table of Contents

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Library of Congress Cataloging-in-Publication Data:

Billingsley, Patrick

Probability and measure / Patrick Billingsley.Anniversary ed.

p. cm.(Wiley series in probability and mathematical statistics. Probability and mathematical statistics)

"A Wiley-Interscience publication."

Includes bibliographical references and index.

ISBN 0-471-00710-2 (3rd. ed.)

ISBN 978-1-118-12237-2 (anniversary ed.)

1. Probabilities. 2. Measure theory. I. Title. II. Series

QA273.B575 1995

519.2dc20

Foreword

Patrick Billingsley, in his Preface to the Third Edition of Probability and Measure wrote that there would not be a fourth edition. More than 15 years have passed since the publication of the third edition, and almost 35 years since the first edition, and yet it remains the standard text and reference for the subject, and it remainsby the standards of graduate-level textbooks in mathematicsa bestseller. Thus, Wiley has decided to reissue the third edition in this new format, the Anniversary Edition of Probability and Measure .

Billingsley was one of the great masters of mathematical exposition and the author of five books on probability and statistics, among which three in particular, Ergodic Theory and Information, Convergence of Probability Measures , and Probability and Measure , have been highly influential, and remain widely cited. Probability and Measure is a classic among mathematical texts. It is unique in its interweaving of measure theory with probability, in the author's words, probability motivating measure theory and measure theory generating further probability. It remains the authoritative treatise on measure-theoretical probability, covering all of the essential bases of the subject in a clear and efficient manner. But it is also a highly individual work, in which the author takes care to show the reader not only the main thoroughfare, but some of the magnificent vistas lying off to the sides. These include brief excursions, many in exercise form, into optimal stopping, the theory of bold play in gambling, probabilistic number theory, and random permutations, among others. There are also side trips into the realm of theoretical statistics, notably the theory of sufficiency and the foundations of Bayesian estimation.

Coming to grips with the measure-theoretical underpinnings of probability can be a daunting task for students of statistics, computer science, and engineering. For a generation of students, Billingsley's book has shown the way up the mountain. This anniversary edition reissue of Probability and Measure ensures that the route will remain open for another generation.

Steve Lalley

Chicago, Illinois

August 2011

Preface

Edward Davenant said he would have a man knockt in the head that should write anything in Mathematiques that had been written of before. So reports John Aubrey in his Brief Lives . What is new here then?

To introduce the idea of measure the book opens with Borel's normal number theorem, proved by calculus alone, and there follow short sections establishing the existence and fundamental properties of probability measures, including Lebesgue measure on the unit interval. For simple random variablesones with finite rangethe expected value is a sum instead of an integral. Measure theory, without integration, therefore suffices for a completely rigorous study of infinite sequences of simple random variables, and this is carried out in the remainder of Chapter 1, which treats laws of large numbers, the optimality of bold play in gambling, Markov chains, large deviations, the law of the iterated logarithm. These developments in their turn motivate the general theory of measure and integration in Chapters 2 and 3.

Measure and integral are used together in Chapters 4 and 5 for the study of random sums, the Poisson process, convergence of measures, characteristic functions, central limit theory. Chapter 6 begins with derivatives according to Lebesgue and RadonNikodyma return to measure theorythen applies them to conditional expected values and martingales. Chapter 7 treats such topics in the theory of stochastic processes as Kolmogorov's existence theorem and separability, all illustrated by Brownian motion.