# An Introduction to Measure-Theoretic Probability

## 2nd Edition

**Authors:**George Roussas

**Hardcover ISBN:**9780128000427

**eBook ISBN:**9780128002902

**Imprint:**Academic Press

**Published Date:**24th March 2014

**Page Count:**426

## Description

*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

## Readership

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

## Table of Contents

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, σ-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

## Details

- No. of pages:
- 426

- Language:
- English

- Copyright:
- © Academic Press 2014

- Published:
- 24th March 2014

- Imprint:
- Academic Press

- Hardcover ISBN:
- 9780128000427

- eBook ISBN:
- 9780128002902

## About the Author

### George Roussas

George G. Roussas earned a B.S. in Mathematics with honors from the University of Athens, Greece, and a Ph.D. in Statistics from the University of California, Berkeley. As of July 2014, he is a Distinguished Professor Emeritus of Statistics at the University of California, Davis. Roussas is the author of five books, the author or co-author of five special volumes, and the author or co-author of dozens of research articles published in leading journals and special volumes. He is a Fellow of the following professional societies: The American Statistical Association (ASA), the Institute of Mathematical Statistics (IMS), The Royal Statistical Society (RSS), the American Association for the Advancement of Science (AAAS), and an Elected Member of the International Statistical Institute (ISI); also, he is a Corresponding Member of the Academy of Athens. Roussas was an associate editor of four journals since their inception, and is now a member of the Editorial Board of the journal Statistical Inference for Stochastic Processes. Throughout his career, Roussas served as Dean, Vice President for Academic Affairs, and Chancellor at two universities; also, he served as an Associate Dean at UC-Davis, helping to transform that institution's statistical unit into one of national and international renown. Roussas has been honored with a Festschrift, and he has given featured interviews for the Statistical Science and the Statistical Periscope. He has contributed an obituary to the IMS Bulletin for Professor-Academician David Blackwell of UC-Berkeley, and has been the coordinating editor of an extensive article of contributions for Professor Blackwell, which was published in the Notices of the American Mathematical Society and the Celebratio Mathematica.

### Affiliations and Expertise

University of California, Davis, USA

## 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**