Real Analysis and Probability
This book serves as a clear, rigorous, and complete introduction to modern probability theory using methods of mathematical analysis, and a description of relations between the two fields. The first half of the book is devoted to an exposition of real analysis. Starting with basic facts of set theory, the book treats e.g. the real number system, transfinite induction, and problems of cardinality, touching both the continuum hypothesis and axiom of choice together with its equivalences. General topology is discussed, including compactness and compactification, completion and completeness, and metric. Measure theory and integration is treated carefully, because it serves as the most important tool for probability theory. Among more advanced topics of real and functional analysis, we can find here an introduction to functional analysis on Banach and Hilbert spaces, convex functions, convex sets and dualities, and measures on topological spaces. The second half of the book contains a description of modern probability theory, including convergence laws, central limit theorems and laws of large numbers. Ergodic theory, as well as martingales, is studied. More advanced topics include convergence laws on separable metric spaces, stochastic processes and Brownian motion. The book can be considered as a textbook. It is a self-contained text and all relevant facts are proved. The appendices show that the author carefully filled all gaps in mathematical background needed later. The book contains a number of exercises helping to understand the contents. It could be very useful for students interested in learning both topics, it can also serve as complementary reading to standard lectures. Teachers preparing their graduate level courses can use the book as an excellent, rigorously written and complete source.