**Mathematical Surveys and Monographs**

Volume: 234;
2018;
286 pp;
Hardcover

MSC: Primary 60; 28; 46; 54;

**Print ISBN: 978-1-4704-4738-0
Product Code: SURV/234**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

**Electronic ISBN: 978-1-4704-4943-8
Product Code: SURV/234.E**

List Price: $122.00

AMS Member Price: $97.60

MAA Member Price: $109.80

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#### Supplemental Materials

# Weak Convergence of Measures

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*Vladimir I. Bogachev*

This book provides a thorough exposition of the main concepts and
results related to various types of convergence of measures arising in
measure theory, probability theory, functional analysis, partial
differential equations, mathematical physics, and other theoretical
and applied fields. Particular attention is given to weak convergence
of measures. The principal material is oriented toward a broad circle
of readers dealing with convergence in distribution of random
variables and weak convergence of measures.

The book contains the necessary background from measure theory and
functional analysis. Large complementary sections aimed at researchers
present the most important recent achievements. More than 100 exercises
(ranging from easy introductory exercises to rather difficult problems
for experienced readers) are given with hints, solutions, or
references. Historic and bibliographic comments are included.

The target readership includes mathematicians and physicists whose
research is related to probability theory, mathematical statistics,
functional analysis, and mathematical physics.

#### Readership

Graduate students and researchers interested in probability theory and functional analysis.

#### Table of Contents

# Table of Contents

## Weak Convergence of Measures

- Cover Cover11
- Title page iii4
- Contents vii8
- Preface ix10
- Chapter 1. Weak convergence of measures on \Reals^{𝑑} 114
- 1.1. Measures and integrals 114
- 1.2. Functions of bounded variation 1023
- 1.3. Facts from functional analysis 1326
- 1.4. Weak convergence of measures on the real line and on \Reals^{𝑑} 2033
- 1.5. Weak convergence of nonnegative measures 2841
- 1.6. Connections with Fourier transforms 3043
- 1.7. Complements and exercises 3851

- Chapter 2. Convergence of measures on metric spaces 4558
- 2.1. Measures on metric spaces 4558
- 2.2. Definition and properties of weak convergence 5164
- 2.3. The Prohorov theorem and weak compactness 5871
- 2.4. Connections with convergence on sets 6275
- 2.5. The case of a Hilbert space 6881
- 2.6. The Skorohod representation 7588
- 2.7. Complements and exercises 7891

- Chapter 3. Metrics on spaces of measures 101114
- Chapter 4. Convergence of measures on topological spaces 139152
- 4.1. Borel, Baire and Radon measures 139152
- 4.2. The weak topology 145158
- 4.3. The case of probability measures 147160
- 4.4. Results of A.D. Alexandroff 154167
- 4.5. Weak compactness 160173
- 4.6. The Fourier transform and weak convergence 167180
- 4.7. Prohorov spaces 171184
- 4.8. Complements and exercises 177190

- Chapter 5. Spaces of measures with the weak topology 199212
- 5.1. Properties of spaces of measures 199212
- 5.2. Mappings of spaces of measures 204217
- 5.3. Continuous inverse mappings 209222
- 5.4. Spaces with the Skorohod property 211224
- 5.5. Uniformly distributed sequences 219232
- 5.6. Setwise convergence of measures 222235
- 5.7. Young measures and the 𝑤𝑠-topology 228241
- 5.8. Complements and exercises 233246

- Comments 245258
- Bibliography 253266
- Index 283296
- Back Cover Back Cover1302