Integration and Probability [Hardcover]

An introduction to analysis with the right mix of abstract theories and concrete problems. Starting with general measure theory, the book goes on to treat Borel and Radon measures and introduces the reader to Fourier analysis in Euclidean spaces with a treatment of Sobolev spaces, distributions, and the corresponding Fourier analysis. It continues with a Hilbertian treatment of the basic laws of probability including Doob's martingale convergence theorem and finishes with Malliavin's stochastic calculus of variations developed in the context of Gaussian measure spaces. This invaluable contribution gives a taste of the fact that analysis is not a collection of independent theories, but can be treated as a whole.This book offers an introduction to analysis with the proper mix of abstract theories and concrete problems, demonstrating for the reader the fact that analysis is not a collection of independent theories but can be treated as a whole.It is a distinct pleasure to have the opportunity to introduce Professor Malliavin's book to the English-speaking mathematical world. In recent years there has been a noticeable retreat from the level of ab? straction at which graduate-level courses in analysis were previously taught in the United States and elsewhere. In contrast to the practices used in the 1950s and 1960s, when great emphasis was placed on the most general context for integration and operator theory, we have recently witnessed an increased emphasis on detailed discussion of integration over Euclidean space and related problems in probability theory, harmonic analysis, and partial differential equations. Professor Malliavin is uniquely qualified to introduce the student to anal? ysis with the proper mix of abstract theories and concrete problems. His mathematical career includes many notable contributions to harmonic anal? ysis, complex analysis, and related problems in probability theory and par? tial differential equations. Rather than developed as a thing-lóä