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.
Author(s): Vladimir I. Bogachev
Series: Mathematical Surveys and Monographs
Publisher: AMS
Year: 2018
Language: English
Pages: 302
Tags: Probability, Measure Theory, Weak Convergence
Cover......Page 1
Title page......Page 4
Contents......Page 8
Preface......Page 10
1.1. Measures and integrals......Page 14
1.2. Functions of bounded variation......Page 23
1.3. Facts from functional analysis......Page 26
1.4. Weak convergence of measures on the real line and on \Reals^{}......Page 33
1.5. Weak convergence of nonnegative measures......Page 41
1.6. Connections with Fourier transforms......Page 43
1.7. Complements and exercises......Page 51
2.1. Measures on metric spaces......Page 58
2.2. Definition and properties of weak convergence......Page 64
2.3. The Prohorov theorem and weak compactness......Page 71
2.4. Connections with convergence on sets......Page 75
2.5. The case of a Hilbert space......Page 81
2.6. The Skorohod representation......Page 88
2.7. Complements and exercises......Page 91
3.1. The weak topology and the Prohorov metric......Page 114
3.2. The Kantorovich and Fortet–Mourier metrics......Page 122
3.3. The Kantorovich metric of order ......Page 130
3.4. Gromov metric triples......Page 135
3.5. Complements and exercises......Page 138
4.1. Borel, Baire and Radon measures......Page 152
4.2. The weak topology......Page 158
4.3. The case of probability measures......Page 160
4.4. Results of A.D. Alexandroff......Page 167
4.5. Weak compactness......Page 173
4.6. The Fourier transform and weak convergence......Page 180
4.7. Prohorov spaces......Page 184
4.8. Complements and exercises......Page 190
5.1. Properties of spaces of measures......Page 212
5.2. Mappings of spaces of measures......Page 217
5.3. Continuous inverse mappings......Page 222
5.4. Spaces with the Skorohod property......Page 224
5.5. Uniformly distributed sequences......Page 232
5.6. Setwise convergence of measures......Page 235
5.7. Young measures and the -topology......Page 241
5.8. Complements and exercises......Page 246
Comments......Page 258
Bibliography......Page 266
Index......Page 296
Back Cover......Page 302
