Praise for the Third Edition "It is, as far as I'm concerned, among the best books in math ever written....if you are a mathematician and want to have the top reference in probability, this is it." (Amazon.com, January 2006) A complete and comprehensive classic in probability and measure theory Probability and Measure, Anniversary Edition by Patrick Billingsley celebrates the achievements and advancements that have made this book a classic in its field for the past 35 years. Now re-issued in a new style and format, but with the reliable content that the third edition was revered for, this Anniversary Edition builds on its strong foundation of measure theory and probability with Billingsley's unique writing style. In recognition of 35 years of publication, impacting tens of thousands of readers, this Anniversary Edition has been completely redesigned in a new, open and user-friendly way in order to appeal to university-level students. This book adds a new foreward by Steve Lally of the Statistics Department at The University of Chicago in order to underscore the many years of successful publication and world-wide popularity and emphasize the educational value of this book. The Anniversary Edition contains features including: * An improved treatment of Brownian motion * Replacement of queuing theory with ergodic theory * Theory and applications used to illustrate real-life situations * Over 300 problems with corresponding, intensive notes and solutions * Updated bibliography * An extensive supplement of additional notes on the problems and chapter commentaries Patrick Billingsley was a first-class, world-renowned authority in probability and measure theory at a leading U.S. institution of higher education. He continued to be an influential probability theorist until his unfortunate death in 2011. Billingsley earned his Bachelor's Degree in Engineering from the U.S. Naval Academy where he served as an officer. he went on to receive his Master's Degree and doctorate in Mathematics from Princeton University.Among his many professional awards was the Mathematical Association of America's Lester R. Ford Award for mathematical exposition. His achievements through his long and esteemed career have solidified Patrick Billingsley's place as a leading authority in the field and been a large reason for his books being regarded as classics. This Anniversary Edition of Probability and Measure offers advanced students, scientists, and engineers an integrated introduction to measure theory and probability. Like the previous editions, this Anniversary Edition is a key resource for students of mathematics, statistics, economics, and a wide variety of disciplines that require a solid understanding of probability theory.
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Table of Contents
Patrick Billingsley: probability theorist and actor, 1925–2011
Chapter 1: Probability
Section 1 Borel's Normal Number Theorem
Section 2 Probability Measures
Section 3 Existence and Extension
Section 4 Denumerable Probabilities
Section 5 Simple Random Variables
Convergence of Random Variables
Section 6 The Law of Large Numbers
Section 7 Gambling Systems
Section 8 Markov Chains
Section 9 Large Deviations and the Law of the Iterated Logarithm
Chapter 2: Measure
Section 10 General Measures
Section 11 Outer Measure
Section 12 Measures in Euclidean Space
Section 13 Measurable Functions and Mappings
Section 14 Distribution Functions
Chapter 3: Integration
Section 15 The Integral
Section 16 Properties Of The Integral
Section 17 The Integral With Respect To Lebesgue Measure
Section 18 Product Measure And Fubini'S Theorem
Section 19 The Lp Spaces
Chapter 4: Random Variables and Expected Values
Section 20 Random Variables and Distributions
Section 21 Expected Values
Section 22 Sums of Independent Random Variables
Section 23 The Poisson Process
Section 24 The Ergodic Theorem
Chapter 5: Convergence of Distributions
Section 25 Weak Convergence
Section 26 Characteristic Functions
Section 27 The Central Limit Theorem
Section 28 Infinitely Divisible Distributions
Section 29 Limit Theorems in Rk
Section 30 The Method of Moments
Chapter 6: Derivatives and Conditional Probability
Section 31 Derivatives on the Line
Section 32 The Radon–Nikodym Theorem
Section 33 Conditional Probability
Section 34 Conditional Expectation
Section 35 Martingales
Chapter 7: Stochastic Processes
Section 36 Kolmogorov's Existence Theorem
Section 37 Brownian Motion
Section 38 Nondenumerable Probabilities
Notes on the Problems
List of Symbols
For further information visit: the book web page http://www.openmodelica.org, the Modelica Association web page http://www.modelica.org, the authors research page http://www.ida.liu.se/labs/pelab/modelica, or home page http://www.ida.liu.se/~petfr/, or email the author at [email protected] Certain material from the Modelica Tutorial and the Modelica Language Specification available at http://www.modelica.org has been reproduced in this book with permission from the Modelica Association under the Modelica License 2 Copyright © 1998–2011, Modelica Association, see the license conditions (including the disclaimer of warranty) at http://www.modelica.org/modelica-legal-documents/ModelicaLicense2.html. Licensed by Modelica Association under the Modelica License 2.
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Copyright © 2011 by the Institute of Electrical and Electronics Engineers, Inc.
Published by John Wiley & Sons, Inc., Hoboken, New Jersey. All rights reserved.
Published simultaneously in Canada.
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Library of Congress Cataloging-in-Publication Data:
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
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 remains—by the standards of graduate-level textbooks in mathematics—a 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.
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 variables—ones with finite range—the 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 Radon–Nikodym—a return to measure theory—then 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.
What is new, then, is the alternation of probability and measure, probability motivating measure theory and measure theory generating further probability. The book presupposes a knowledge of combinatorial and discrete probability, of rigorous calculus, in particular infinite series, and of elementary set theory. Chapters 1 through 4 are designed to be taken up in sequence. Apart from starred sections and some examples, Chapter 5, 6, and 7 are independent of one another; they can be read in any order.
My goal has been to write a book I would myself have liked when I first took up the subject, and the needs of students have been given precedence over the requirements of logical economy. For instance, Kolmogorov's existence theorem appears not in the first chapter but in the last, stochastic processes needed earlier having been constructed by special arguments which, although technically redundant, motivate the general result. And the general result is, in the last chapter, given two proofs at that. It is instructive, I think, to see the show in rehearsal as well as in performance.
The Third Edition. The main changes in this edition are two. For the theory of Hausdorff measures in Section 19 I have substituted an account of L p spaces, with applications to statistics. And for the queueing theory in Section 24 I have substituted an introduction to ergodic theory, with applications to continued fractions and Diophantine approximation. These sections now fit better with the rest of the book, and they illustrate again the connections probability theory has with applied mathematics on the one hand and with pure mathematics on the other.
For suggestions that have led to improvements in the new edition, I thank Raj Bahadur, Walter Philipp, Michael Wichura, and Wing Wong, as well as the many readers who have sent their comments.
Envoy. I said in the preface to the second edition that there would not be a third, and yet here it is. There will not be a fourth. It has been a very agreeable labor, writing these successive editions of my contribution to the river of mathematics. And although the contribution is small, the river is great: After ages of good service done to those who people its banks, as Joseph Conrad said of the Thames, it spreads out “in the tranquil dignity of a waterway leading to the uttermost ends of the earth.”
Patrick Billingsley: probability theorist and actor, 1925–2011
Patrick Billingsley, professor emeritus in statistics and mathematics
By Steve Koppes
(This article was originally published April 29, 2011, in UCHICAGO NEWS.)
Patrick Billingsley was an influential probability theorist who also became an accomplished actor of stage and “He's most known for his series of books in advanced probability theory,” said Steve Lalley, UChicago professor in statistics. “They are all models of exposition. They really are fine works of mathematical writing. Several generations of graduate students in both probability and statistics have learned their basic probability from these books. They continue to be used, and they continue to be cited.”
Billingsley, professor emeritus in statistics and mathematics, first took the stage in fifth grade, when he played Robin Hood, according to a 1970 article published in the Chicago Maroon. He later performed in “The Revels,” the annual faculty review.
He began acting in earnest in 1966 at the University's Court Theatre, back when the professional company was an amateur company staging open-air plays in Hutchinson Courtyard.
Billingsley held leading roles in more than 20 productions at Court Theatre and Body Politic Theatre in Chicago. His roles included the Captain in We Bombed in New Haven (1970); Alonzo in The Tempest (1977); Dysart in Equus, (1980); and Petey in The Birthday Party (1978 and 1985).
A talent scout saw Billingsley perform in The Lover in 1977, which led to his successful audition for a part in the 1978 Kirk Douglas film The Fury. Billingsley never met Douglas, but they appear on screen together during a car chase on Wacker Drive and Van Buren Street in Chicago. Billingsley played a bad guy who ended up dying in a fiery crash.
Billingsley went on to appear in seven more films and in nine television shows. His movie roles included playing a biology teacher in My Bodyguard (1980), the professor in Somewhere in Time (1980) and the bailiff in The Untouchables (1987).
“When you teach, you perform in front of an audience. That's much like acting. As a teacher you're used to being on stage,” Billingsley told the Chicago Tribune Magazine in 1978.
Family members and friends knew Billingsley as a man with a zest for life, good cheer and a wry sense of humor, said his daughter Marty Billingsley, a teacher at UChicago's Laboratory Schools. “He was the type of guy who read Mad Magazine along with The New Yorker and watched ‘Monty Python’ as well as the PBS ‘NewsHour’ on TV,” she said.
Her father lived a life of both mind and body, she noted. He worked out daily in the Henry Crown Field House on campus for 40 years, served on the athletic board and even helped run the football scoreboard.
“A true Renaissance man, he also painted, did woodworking, sung Child Ballads as lullabies to his children—echoes of which have made their way into his daughter Franny Billingsley's young-adult novels—and read Beowulf in the original Old English,” Marty Billingsley said.
Billingsley was born May 3, 1925 in Sioux Falls, S.D. He earned a bachelor's degree in engineering from the U.S. Naval Academy in 1948, then served in the U.S. Navy until 1957. As a Navy officer he lived for a year in Japan, where he earned a black belt in judo.
He attended Princeton University for graduate studies and received a master's degree in 1952 and a doctorate in 1955, both in mathematics. Billingsley worked as a National Science Foundation Fellow in Mathematics at Princeton in 1957–58.
Billingsley joined the UChicago faculty as an assistant professor in statistics in 1958, attaining the rank of professor in statistics and mathematics five years later. He served as department chairman from 1980 to 1983 and retired as professor emeritus in 1994.
A member of the American Academy of Arts and Sciences and a fellow of the Institute of Mathematical Statistics, his honors also include the Mathematical Association of America's Lester R. Ford award for mathematical exposition.
Billingsley was a Fulbright Fellow and visiting professor at the University of Copenhagen, Denmark (1964–65) and a Guggenheim Fellow and visiting professor at the University of Cambridge, England, in 1971–72. He served as editor of the Annals of Probability from 1976 to 1979 and as president of the Institute of Mathematical Statistics in 1983.
He was the author or co-author of five books, including Statistical Inference for Markov Processes (1961), Ergodic Theory and Information (1965), Convergence of Probability Measures (1968), The Elements of Statistical Inference (1986). His Probability and Measure (1986) was translated into Polish.
He delivered numerous lectures internationally, addressing students and colleagues in England, India, Scotland, Sweden, and Italy. In 1970 he gave an invited address at the American Mathematical Society's annual meetings.
Billingsley is survived by his children Franny, Patty, Julie, Marty and Paul, who is an assistant project manager with UChicago's Facilities Services; and by his companion, Florence Weisblatt. His late wife of nearly 50 years, social activist Ruth Billingsley, died in 2000.
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