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An Introduction to Measure-Theoretic Probability,

Edition 2

Editors: By George G. Roussas

Publication Date: 24 Mar 2014

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An Introduction to Measure-Theoretic Probability, Second Edition, employs a classical approach to teaching the basics of measure theoretic probability. This book provides in a concise, yet detailed way, the bulk of the probabilistic tools that a student working toward an advanced degree in statistics, probability and other related areas should be equipped with.

This edition requires no prior knowledge of measure theory, covers all its topics in great detail, and includes one chapter on the basics of ergodic theory and one chapter on two cases of statistical estimation. Topics range from the basic properties of a measure to modes of convergence of a sequence of random variables and their relationships; the integral of a random variable and its basic properties; standard convergence theorems; standard moment and probability inequalities; the Hahn-Jordan Decomposition Theorem; the Lebesgue Decomposition T; conditional expectation and conditional probability; theory of characteristic functions; sequences of independent random variables; and ergodic theory. There is a considerable bend toward the way probability is actually used in statistical research, finance, and other academic and nonacademic applied pursuits. Extensive exercises and practical examples are included, and all proofs are presented in full detail. Complete and detailed solutions to all exercises are available to the instructors on the book companion site.

This text will be a valuable resource for graduate students primarily in statistics, mathematics, electrical and computer engineering or other information sciences, as well as for those in mathematical economics/finance in the departments of economics.

Key Features

Provides in a concise, yet detailed way, the bulk of probabilistic tools essential to a student working toward an advanced degree in statistics, probability, and other related fields
Includes extensive exercises and practical examples to make complex ideas of advanced probability accessible to graduate students in statistics, probability, and related fields
All proofs presented in full detail and complete and detailed solutions to all exercises are available to the instructors on book companion site
Considerable bend toward the way probability is used in statistics in non-mathematical settings in academic, research and corporate/finance pursuits

About the author

By George G. Roussas, University of California, Davis, USA

Pictured on the Cover

Carathéodory, Constantine (1873–1950)

Preface to First Edition

Preface to Second Edition

Chapter 1. Certain Classes of Sets, Measurability, and Pointwise Approximation

Abstract
1.1 Measurable Spaces
1.2 Product Measurable Spaces
1.3 Measurable Functions and Random Variables

Chapter 2. Definition and Construction of a Measure and its Basic Properties

Abstract
2.1 About Measures in General, and Probability Measures in Particular
2.2 Outer Measures
2.3 The Carathéodory Extension Theorem
2.4 Measures and (Point) Functions

Chapter 3. Some Modes of Convergence of Sequences of Random Variables and their Relationships

Abstract
3.1 Almost Everywhere Convergence and Convergence in Measure
3.2 Convergence in Measure is Equivalent to Mutual Convergence in Measure

Chapter 4. The Integral of a Random Variable and its Basic Properties

Abstract
4.1 Definition of the Integral
4.2 Basic Properties of the Integral
4.3 Probability Distributions

Chapter 5. Standard Convergence Theorems, The Fubini Theorem

Abstract
5.1 Standard Convergence Theorems and Some of Their Ramifications
5.2 Sections, Product Measure Theorem, the Fubini Theorem

Chapter 6. Standard Moment and Probability Inequalities, Convergence in the rth Mean and its Implications

Abstract
6.1 Moment and Probability Inequalities
6.2 Convergence in the rth Mean, Uniform Continuity, Uniform Integrability, and their Relationships

Chapter 7. The Hahn–Jordan Decomposition Theorem, The Lebesgue Decomposition Theorem, and the Radon–Nikodym Theorem

Abstract
7.1 The Hahn–Jordan Decomposition Theorem
7.2 The Lebesgue Decomposition Theorem
7.3 The Radon–Nikodym Theorem

Chapter 8. Distribution Functions and Their Basic Properties, Helly–Bray Type Results

Abstract
8.1 Basic Properties of Distribution Functions
8.2 Weak Convergence and Compactness of a Sequence of Distribution Functions
8.3 Helly–Bray Type Theorems for Distribution Functions

Chapter 9. Conditional Expectation and Conditional Probability, and Related Properties and Results

Abstract
9.1 Definition of Conditional Expectation and Conditional Probability
9.2 Some Basic Theorems About Conditional Expectations and Conditional Probabilities
9.3 Convergence Theorems and Inequalities for Conditional Expectations
9.4 Further Properties of Conditional Expectations and Conditional Probabilities

Chapter 10. Independence

Abstract
10.1 Independence of Events, s-Fields, and Random Variables
10.2 Some Auxiliary Results
10.3 Proof of Theorem 1 and of Lemma 1 in Chapter 9

Chapter 11. Topics from the Theory of Characteristic Functions

Abstract
11.1 Definition of the Characteristic Function of a Distribution and Basic Properties
11.2 The Inversion Formula
11.3 Convergence in Distribution and Convergence of Characteristic Functions—The Paul Lévy Continuity Theorem
11.4 Convergence in Distribution in the Multidimensional Case—The Cramér–Wold Device
11.5 Convolution of Distribution Functions and Related Results
11.6 Some Further Properties of Characteristic Functions
11.7 Applications to the Weak Law of Large Numbers and the Central Limit Theorem
11.8 The Moments of a Random Variable Determine its Distribution
11.9 Some Basic Concepts and Results from Complex Analysis Employed in the Proof of Theorem 11

Chapter 12. The Central Limit Problem: The Centered Case

Abstract
12.1 Convergence to the Normal Law (Central Limit Theorem, CLT)
12.2 Limiting Laws of L(Sn) Under Conditions (C)
12.3 Conditions for the Central Limit Theorem to Hold
12.4 Proof of Results in Section 12.2

Chapter 13. The Central Limit Problem: The Noncentered Case

Abstract
13.1 Notation and Preliminary Discussion
13.2 Limiting Laws of L(Sn) Under Conditions (C¿)
13.3 Two Special Cases of the Limiting Laws of L(Sn)

Chapter 14. Topics from Sequences of Independent Random Variables

Abstract
14.1 Kolmogorov Inequalities
14.2 More Important Results Toward Proving the Strong Law of Large Numbers
14.3 Statement and Proof of the Strong Law of Large Numbers
14.4 A Version of the Strong Law of Large Numbers for Random Variables with Infinite Expectation
14.5 Some Further Results on Sequences of Independent Random Variables

Chapter 15. Topics from Ergodic Theory

Abstract
15.1 Stochastic Process, the Coordinate Process, Stationary Process, and Related Results
15.2 Measure-Preserving Transformations, the Shift Transformation, and Related Results
15.3 Invariant and Almost Sure Invariant Sets Relative to a Transformation, and Related Results
15.4 Measure-Preserving Ergodic Transformations, Invariant Random Variables Relative to a Transformation, and Related Results
15.5 The Ergodic Theorem, Preliminary Results
15.6 Invariant Sets and Random Variables Relative to a Process, Formulation of the Ergodic Theorem in Terms of Stationary Processes, Ergodic Processes

Chapter 16. Two Cases of Statistical Inference: Estimation of a Real-Valued Parameter, Nonparametric Estimation of a Probability Density Function

Abstract
16.1 Construction of an Estimate of a Real-Valued Parameter
16.2 Construction of a Strongly Consistent Estimate of a Real-Valued Parameter
16.3 Some Preliminary Results
16.4 Asymptotic Normality of the Strongly Consistent Estimate
16.5 Nonparametric Estimation of a Probability Density Function
16.6 Proof of Theorems 3–5

Appendix A. Brief Review of Chapters 1–16

Chapter 1 Certain Classes of Sets, Measurability, and Pointwise Approximation
Chapter 2 Definition and Construction of a Measure and its Basic Properties
Chapter 3 Some Modes of Convergence of Sequences of Random Variables and their Relationships
Chapter 4 The Integral of a Random Variable and its Basic Properties
Chapter 5 Standard Convergence Theorems, The Fubini Theorem
Chapter 6 Standard Moment and Probability Inequalities, Convergence in the rth Mean and its Implications
Chapter 7 The Hahn–Jordan Decomposition Theorem, The Lebesgue Decomposition Theorem, and the Radon–Nikodym Theorem
Chapter 8 Distribution Functions and Their Basic Properties, Helly–Bray Type Results
Chapter 9 Conditional Expectation and Conditional Probability, and Related Properties and Results
Chapter 10 Independence
Chapter 11 Topics from the Theory of Characteristic Functions
Chapter 12 The Central Limit Problem: The Centered Case
Chapter 13 The Central Limit Problem: The Non-centered Case
Chapter 14 Topics from Sequences of Independent Random Variables
Chapter 15 Topics from Ergodic Theory
Chapter 16 Two Cases of Statistical Inference: Estimation of a Real-valued Parameter, Nonparametric Estimation of a Probability Density Function

Appendix B. Brief Review of Riemann–Stieltjes Integral

Appendix C. Notation and Abbreviations

Selected References

Revised Answers Manual to an Introduction to Measure-Theoretic Probability

Chapter 1
Chapter 2
Chapter 3
Chapter 4
Chapter 5
Chapter 6
Chapter 7
Chapter 8
Chapter 9
Chapter 10
Chapter 11
Chapter 12
Chapter 13
Chapter 14
Chapter 15

Title Reviews

"...a very thorough discussion of many of the pillars of the subject, showing in particular how 'measure theory with total measure one' is just the tip of the iceberg...It’s quite a book."—MAA.org, An Introduction to Measure-Theoretic Probability

"This second edition employs a classical approach to teaching students of statistics, mathematics, engineering, econometrics, finance, and other disciplines measure-theoretic probability…requires no prior knowledge of measure theory, discusses all its topics in great detail, and includes one chapter on the basics of ergodic theory and one chapter on two cases of statistical estimation."—Zentralblatt MATH 1287-1

"...provides basic tools in measure theory and probability, in the classical spirit, relying heavily on characteristic functions as tools without using martingale or empirical process methods. A well-written book. Highly recommended [for] graduate students; faculty."—CHOICE

Based on the material presented in the manuscript, I would without any hesitation adopt the published version of the book. The topics dealt are essential to the understanding of more advanced material; the discussion is deep and it is combined with the use of essential technical details. It will be an extremely useful book. In addition it will be a very popular book."—Madan Puri, Indiana University

"Would likely use as one of two required references when I teach either Stat 709 or Stat 732 again. Would also highly recommend to colleagues. The author has written other excellent graduate texts in mathematical statistics and contiguity and this promises to be another. This book could well become an important reference for mathematical statisticians."—Richard Johnson, University of Wisconsin

"The author has succeeded in making certain deep and fundamental ideas of probability and measure theory accessible to statistics majors heading in the direction of graduate studies in statistical theory."—Doraiswamy Ramachandran, California State University

ISBN: 9780128000427

Page Count: 426

Retail Price (USD) :

Roussas: An Introduction to Probability and Statistical Inference $135.00; 2002, ISBN 9780125990202

Roussas: An Introduction to Measure-theoretic Probability $137.00; 2004, ISBN 9780125990226

Smith: Essential Statistics, Regression, and Econometrics $94.95; 2011, ISBN 9780123822215

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Graduate students primarily in statistics, mathematics, electrical & computer engineering or other information sciences; mathematical economics/finance in departments of economics.

An Introduction to Measure-Theoretic Probability,

Editors: By George G. Roussas

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About the author

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