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This book will enable researchers and students of analysis to more easily understand research papers in which probabilistic methods are used to prove theorems of analysis, many of which have no other known proofs. The book assumes a course in measure and integration theory but requires little or no background in probability theory. It emplhasizes topics of interest to analysts, including random series, martingales and Brownian motion.

This book is first of all designed as a text for the course usually called "theory of functions of a real variable." This course is at present cus tomarily otIered as a first or second year graduate course in United States universities, although there are signs that this sort of analysis will soon penetrate upper division undergraduate curricula. We have inc1uded every topic that we think essential for the training of analysts, and we have also gone down a number of interesting bypaths. We hope too that the book will be useful as a reference for mature mathematicians and other scientific...