Integration and Probability

16This book is designed to be an introduction to analysis with the proper mix of abstract theories and concrete problems. It starts with general measure theory, treats Borel and Radon measures (with particular attention paid to Lebesgue measure) and introduces the reader to Fourier analysis in Euclidean spaces with a treatment of Sobolev spaces, distributions, and the Fourier analysis of such. 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 to the existing literature gives the reader a taste of the fact that analysis is not a collection of independent theories but can be treated as a whole.04BExtract of ContentsI. Measurable Spaces and Integrable FunctionsII. Borel Measures and Radon MeasuresIII. Fourier AnalysisIV. Hilbert Space Methods and Limit Theorems in Probability TheoryV. Gaussian Sobolev Spaces and Stochastic Calculus of VariationsAppendix I. Hilbert Spectral AnalysisAppendix II. Infinitesimal and Integrated Forms of the Change-of-Variables Formula