# Probability and Measure, Anniversary Edition

# Probability and Measure, Anniversary Edition

ISBN: 978-1-118-12237-2 February 2012 656 Pages

## Description

**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

PREFACE xiii

*Patrick Billingsley* 1925–2011 xv

**Chapter1 PROBABILITY 1**

**1. BOREL’S NORMAL NUMBER THEOREM, 1**

The Unit Interval

The Weak Law of Large Numbers

The Strong Law of Large Numbers

Strong Law Versus Weak

Length

The Measure Theory of Diophantine Approximation*

**2. PROBABILITY MEASURES**, **18**

Spaces

Assigning Probabilities

Classes of Sets

Probability Measures

Lebesgue Measure on the Unit Interval

Sequence Space*

Constructing *s*-Fields*

**3. EXISTENCE AND EXTENSION**, **39**

Construction of the Extension

Uniqueness and the *p***–***?* Theorem

Monotone Classes

Lebesgue Measure on the Unit Interval

Completeness

Nonmeasurable Sets

Two Impossibility Theorems*

**4. DENUMERABLE PROBABILITIES**, **53**

General Formulas

Limit Sets

Independent Events

Subfields

The Borel-Cantelli Lemmas

The Zero-One Law

**5. SIMPLE RANDOM VARIABLES**, **72**

Definition

Convergence of Random Variables

Independence

Existence of Independent Sequences

Expected Value

Inequalities

**6. THE LAW OF LARGE NUMBERS**, **90**

The Strong Law

The Weak Law

Bernstein's Theorem

A Refinement of the Second Borel-Cantelli Lemma

**7. GAMBLING SYSTEMS**, **98**

Gambler's Ruin

Selection Systems

Gambling Policies

Bold Play*

Timid Play*

**8. MARKOV CHAINS**, **117**

Definitions

Higher-Order Transitions

An Existence Theorem

Transience and Persistence

Another Criterion for Persistence

Stationary Distributions

Exponential Convergence*

Optimal Stopping*

**9. LARGE DEVIATIONS AND THE LAW OF THE ITERATED LOGARITHM**, **154**

Moment Generating Functions

Large Deviations

Chernoff's Theorem*

The Law of the Iterated Logarithm

**Chapter2 MEASURE 167**

**10. GENERAL MEASURES, 167**

Classes of Sets

Conventions Involving 8

Measures

Uniqueness

**11. OUTER MEASURE**, **174**

Outer Measure

Extension

An Approximation Theorem

**12. MEASURES IN EUCLIDEAN SPACE**, **181**

Lebesgue Measure

Regularity

Specifying Measures on the Line

Specifying Measures in *Rk*

Strange Euclidean Sets*

**13. MEASURABLE FUNCTIONS AND MAPPINGS**, **192**

Measurable Mappings

Mappings into *Rk*

Limits and Measurability

Transformations of Measures

**14. DISTRIBUTION FUNCTIONS**, **198**

Distribution Functions

Exponential Distributions

Weak Convergence

Convergence of Types*

Extremal Distributions*

**Chapter3 INTEGRATION 211**

**15. THE INTEGRAL, 211**

Definition

Nonnegative Functions

Uniqueness

**16. PROPERTIES OF THE INTEGRAL**, **218**

Equalities and Inequalities

Integration to the Limit

Integration over Sets

Densities

Change of Variable

Uniform Integrability

Complex Functions

**17. THE INTEGRAL WITH RESPECT TO LEBESGUE MEASURE**, **234**

The Lebesgue Integral on the Line

The Riemann Integral

The Fundamental Theorem of Calculus

Change of Variable

The Lebesgue Integral in *Rk*

Stieltjes Integrals

**18. PRODUCT MEASURE AND FUBINI’S THEOREM**, **245**

Product Spaces

Product Measure

Fubini's Theorem

Integration by Parts

Products of Higher Order

**19. THE Lp SPACES***,

**256**

Definitions

Completeness and Separability

Conjugate Spaces

Weak Compactness

Some Decision Theory

The Space *L*2

An Estimation Problem

**Chapter4 RANDOM VARIABLES AND EXPECTED VALUES 271**

**20. RANDOM VARIABLES AND DISTRIBUTIONS, 271**

Random Variables and Vectors

Subfields

Distributions

Multidimensional Distributions

Independence

Sequences of Random Variables

Convolution

Convergence in Probability

The Glivenko-Cantelli Theorem*

**21. EXPECTED VALUES**, **291**

Expected Value as Integral

Expected Values and Limits

Expected Values and Distributions

Moments

Inequalities

Joint Integrals

Independence and Expected Value

Moment Generating Functions

**22. SUMS OF INDEPENDENT RANDOM VARIABLES**, **300**

The Strong Law of Large Numbers

The Weak Law and Moment Generating Functions

Kolmogorov's Zero-One Law

Maximal Inequalities

Convergence of Random Series

Random Taylor Series*

**23. THE POISSON PROCESS**, **316**

Characterization of the Exponential Distribution

The Poisson Process

The Poisson Approximation

Other Characterizations of the Poisson Process

Stochastic Processes

**24. THE ERGODIC THEOREM***, **330**

Measure-Preserving Transformations

Ergodicity

Ergodicity of Rotations

Proof of the Ergodic Theorem

The Continued-Fraction Transformation

Diophantine Approximation

**Chapter5 CONVERGENCE OF DISTRIBUTIONS 349**

**25. WEAK CONVERGENCE, 349**

Definitions

Uniform Distribution Modulo 1*

Convergence in Distribution

Convergence in Probability

Fundamental Theorems

Helly's Theorem

Integration to the Limit

**26. CHARACTERISTIC FUNCTIONS**, **365**

Definition

Moments and Derivatives

Independence

Inversion and the Uniqueness Theorem

The Continuity Theorem

Fourier Series*

**27. THE CENTRAL LIMIT THEOREM**, **380**

Identically Distributed Summands

The Lindeberg and Lyapounov Theorems

Dependent Variables*

**28. INFINITELY DIVISIBLE DISTRIBUTIONS***, **394**

Vague Convergence

The Possible Limits

Characterizing the Limit

**29. LIMIT THEOREMS IN Rk**,

**402**

The Basic Theorems

Characteristic Functions

Normal Distributions in *Rk*

The Central Limit Theorem

**30. THE METHOD OF MOMENTS***, **412**

The Moment Problem

Moment Generating Functions

Central Limit Theorem by Moments

Application to Sampling Theory

Application to Number Theory

**Chapter6 DERIVATIVES AND CONDITIONAL PROBABILITY 425**

**31. DERIVATIVES ON THE LINE*, 425**

The Fundamental Theorem of Calculus

Derivatives of Integrals

Singular Functions

Integrals of Derivatives

Functions of Bounded Variation

**32. THE RADON–NIKODYM THEOREM**, **446**

Additive Set Functions

The Hahn Decomposition

Absolute Continuity and Singularity

The Main Theorem

**33. CONDITIONAL PROBABILITY**, **454**

The Discrete Case

The General Case

Properties of Conditional Probability

Difficulties and Curiosities

Conditional Probability Distributions

**34. CONDITIONAL EXPECTATION**, **472**

Definition

Properties of Conditional Expectation

Conditional Distributions and Expectations

Sufficient Subfields*

Minimum-Variance Estimation*

**35. MARTINGALES**, **487**

Definition

Submartingales

Gambling

Functions of Martingales

Stopping Times

Inequalities

Convergence Theorems

Applications: Derivatives

Likelihood Ratios

Reversed Martingales

Applications: de Finetti's Theorem

Bayes Estimation

A Central Limit Theorem*

**Chapter7 STOCHASTIC PROCESSES 513**

**36. KOLMOGOROV'S EXISTENCE THEOREM, 513**

Stochastic Processes

Finite-Dimensional Distributions

Product Spaces

Kolmogorov's Existence Theorem

The Inadequacy of *RT*

A Return to Ergodic Theory

The Hewitt**–**Savage Theorem*

**37. BROWNIAN MOTION**, **530**

Definition

Continuity of Paths

Measurable Processes

Irregularity of Brownian Motion Paths

The Strong Markov Property

The Reflection Principle

Skorohod Embedding

Invariance*

**38. NONDENUMERABLE PROBABILITIES**, **558**

Introduction

Definitions

Existence Theorems

Consequences of Separability*

APPENDIX 571

NOTES ON THE PROBLEMS 587

BIBLIOGRAPHY 617

INDEX 619

## Reviews

“Like the previous editions, this Anniversary edition will be well received by students of mathematics, statistics, economics, and a wide variety of disciplines that require a solid understanding of probability theory.” (*Int. J. Microstructure and Materials Properties*, 1 February 2013)