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Foundations of Stochastic Analysis deals with the foundations of the theory of Kolmogorov and Bochner and its impact on the growth of stochastic analysis. Topics covered range from conditional expectations and probabilities to projective and direct limits, as well as martingales and likelihood ratios. Abstract martingales and their applications are also discussed. Comprised of five chapters, this volume begins with an overview of the basic Kolmogorov-Bochner theorem, followed by a discussion on conditional expectations and probabilities containing several characterizations of operators and measures. The applications of these conditional expectations and probabilities to Reynolds operators are also considered. The reader is then introduced to projective limits, direct limits, and a generalized Kolmogorov existence theorem, along with infinite product conditional probability measures. The book also considers martingales and their applications to likelihood ratios before concluding with a description of abstract martingales and their applications to convergence and harmonic analysis, as well as their relation to ergodic theory. This monograph should be of considerable interest to researchers and graduate students working in stochastic analysis. Content: Front Matter, Page iii Copyright, Page iv Dedication, Page v Preface, Pages ix-xi CHAPTER I - Introduction and Generalities, Pages 1-36 CHAPTER II - Conditional Expectations and Probabilities, Pages 37-115 CHAPTER III - Projective and Direct Limits, Pages 116-169 CHAPTER IV - Martingales and Likelihood Ratios, Pages 170-240 CHAPTER V - Abstract Martingales and Applications, Pages 241-282 Bibliography, Pages 283-290 Index, Pages 291-295 Probability and Mathematical Statistics: A Series of Monographs and Textbooks, Pages ibc1-ibc2 M.m. Rao. Includes Index. Bibliography: P. 283-290.