Real Analysis and Probability (Probability and mathematical statistics; a series of monographs and textbooks)

Real Analysis and Probability provides the background in real analysis needed for the study of probability. Topics covered range from measure and integration theory to functional analysis and basic concepts of probability. The interplay between measure theory and topology is also discussed, along with conditional probability and expectation, the central limit theorem, and strong laws of large numbers with respect to martingale theory. Comprised of eight chapters, this volume begins with an overview of the basic concepts of the theory of measure and integration, followed by a presentation of various applications of the basic integration theory. The reader is then introduced to functional analysis, with emphasis on structures that can be defined on vector spaces. Subsequent chapters focus on the connection between measure theory and topology; basic concepts of probability; and conditional probability and expectation. Strong laws of large numbers are also examined, first from the classical viewpoint, and then via martingale theory. The final chapter is devoted to the one-dimensional central limit problem, paying particular attention to the fundamental role of Prokhorovs weak compactness theorem. This book is intended primarily for students taking a graduate course in probability. Real Analysis and Solutions to Problems presents solutions to problems in real analysis and probability. Topics covered range from measure and integration theory to functional analysis and basic concepts of probability; the interplay between measure theory and topology; conditional probability and expectation; the central limit theorem; and strong laws of large numbers in terms of martingale theory.Comprised of eight chapters, this volume begins with problems and solutions for the theory of measure and integration, followed by various applications of the basic integration theory. Subsequent chapters deal with functional analysis, paying particular attention to structures that can be defined on vector spaces; the connection between measure theory and topology; basic concepts of probability; and conditional probability and expectation. Strong laws of large numbers are also taken into account, first from the classical viewpoint, and then via martingale theory. The final chapter is devoted to the one-dimensional central limit problem, with emphasis on the fundamental role of Prokhorov's weak compactness theorem.This book is intended primarily for students taking a graduate course in probability. Content: Front Matter, Page iii Copyright, Page iv Preface, Pages ix-x Summary of Notation, Pages xi-xv 1 - Fundamentals of Measure and Integration Theory, Pages 1-57 2 - Further Results in Measure and Integration Theory, Pages 58-112 3 - Introduction to Functional Analysis, Pages 113-167 4 - The Interplay between Measure Theory and Topology, Pages 168-200 5 - Basic Concepts of Probability, Pages 201-235 6 - Conditional Probability and Expectation, Pages 236-268 7 - Strong Laws of Large Numbers and Martingale Theory, Pages 269-320 8 - The Central Limit Theorem, Pages 321-367 Appendix on General Topology, Pages 369-408 Bibliography, Page 409 Solutions to Problems, Pages 411-467 Subject Index, Pages 469-476 Probability and Mathematical Statistics: A Series of Monographs and Textbooks, Pages ibc1-ibc2