# Probability Measures on Metric Spaces

## 1st Edition

**Authors:**K. R. Parthasarathy

**Editors:**Z. W. Birnbaum E. Lukacs

**eBook ISBN:**9781483225258

**Imprint:**Academic Press

**Published Date:**1st January 1967

**Page Count:**288

## Description

Probability Measures on Metric Spaces presents the general theory of probability measures in abstract metric spaces. This book deals with complete separable metric groups, locally impact abelian groups, Hilbert spaces, and the spaces of continuous functions.

Organized into seven chapters, this book begins with an overview of isomorphism theorem, which states that two Borel subsets of complete separable metric spaces are isomorphic if and only if they have the same cardinality. This text then deals with properties such as tightness, regularity, and perfectness of measures defined on metric spaces. Other chapters consider the arithmetic of probability distributions in topological groups. This book discusses as well the proofs of the classical extension theorems and existence of conditional and regular conditional probabilities in standard Borel spaces. The final chapter deals with the compactness criteria for sets of probability measures and their applications to testing statistical hypotheses.

This book is a valuable resource for statisticians.

## Table of Contents

Preface

Chapter I—The Borel Subsets of a Metric Space

1. General Properties of Borel Sets

2. The Isomorphism Theorem

3. The Kuratowski Theorem

4. Borel Cross Sections in Compact Metric Spaces

5. Borel Cross Sections in Locally Compact Groups

Chapter II—Probability Measures in a Metric Space

1. Regular Measures

2. Spectrum of a Measure

3. Tight Measures

4. Perfect Measures

5. Linear Functionals and Measures

6. The Weak Topology in the Space of Measures

7. Convergence of Sample Distributions

8. Existence of Nonatomic Measures in Metric Spaces

Chapter III—Probability Measures in a Metric Group

1. The Convolution Operation

2. Shift Compactness in ℳ(X)

3. Idempotent Measures

4. Indecomposable Measures

5. The Case When X Is Abelian

Chapter IV—Probability Measures in Locally Compact Abelian Groups

1. Introduction

2. Preliminary Facts About a Group and Its Character Group

3. Measures and Their Fourier Transforms

4. Infinitely Divisible Distributions

5. General Limit Theorems for Sums of Infinitesimal Summands

6. Gaussian Distributions

7. Representation of Infinitely Divisible Distributions

8. Uniqueness of the Representation

9. Compactness Criteria

10. Representation of Convolution Semigroups

11. A Decomposition Theorem

12. Absolutely Continuous Indecomposable Distributions in X

Chapter V—The Kolmogorov Consistency Theorem and Conditional Probability

1. Statement of the First Problem

2. Standard Borel Spaces

3. The Consistency Theorem in the Case of Inverse Limits of Borel Spaces

4. The Extension Theorem

5. The Kolmogorov Consistency Theorem

6. Statement of the Second Problem

7. Existence of Conditional Probability

8. Regular Conditional Probability

Chapter VI—Probability Measures in a Hilbert Space

1. Introduction

2. Characteristic Functions and Compactness Criteria

3. An Estimate of the Variance

4. Infinitely Divisible Distributions

5. Compactness Criteria

6. Accompanying Laws

7. Representation of Convolution Semigroups

8. Decomposition Theorem

9. Ergodic Theorems

Chapter VII—Probability Measures on C[0, 1] and D[0, 1]

1. Introduction

2. Probability Measures on C[0, 1]

3. A Condition for the Realization of a Stochastic Process in "C"

4. Convergence to Brownian Motion

5. Distributions of Certain Random Variables Associated with the Brownian Motion

6. The Space D[0, 1]

7. Probability Measures in "D"

8. Ergodic Theorems for D-Valued Random Variables

9. Applications to Statistical Tests of Hypothesis

Bibliographical Notes

Bibliography

List of Symbols

Author Index

Subject Index

## Details

- No. of pages:
- 288

- Language:
- English

- Copyright:
- © Academic Press 1967

- Published:
- 1st January 1967

- Imprint:
- Academic Press

- eBook ISBN:
- 9781483225258

## About the Author

### K. R. Parthasarathy

## About the Editor

### Z. W. Birnbaum

### E. Lukacs

### Affiliations and Expertise

Bowling Green State University