*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

**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 *r*th Mean and its Implications

- Abstract
- 6.1 Moment and Probability Inequalities
- 6.2 Convergence in the
*r*th 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

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