Modern Analysis (1997): (CRC Press Revivals)

Modern Analysis provides coverage of real and abstract analysis, offering a sensible introduction to functional analysis as well as a thorough discussion of measure theory, Lebesgue integration, and related topics. This significant study clearly and distinctively presents the teaching and research literature of graduate analysis: Providing a fundamental, modern approach to measure theory Investigating advanced material on the Bochner integral, geometric theory, and major theorems in Fourier Analysis Rn, including the theory of singular integrals and Milhin's theorem - material that does not appear in textbooks Offering exceptionally concise and cardinal versions of all the main theorems about characteristic functions Containing an original examination of sufficient statistics, based on the general theory of Radon measures With an ambitious scope, this resource unifies various topics into one volume succinctly and completely. The contents span basic measure theory in an abstract and concrete form, material on classic linear functional analysis, probability, and some major results used in the theory of partial differential equations. Two different proofs of the central limit theorem are examined as well as a straightforward approach to conditional probability and expectation. Modern Analysis provides ample and well-constructed exercises and examples. Introductory topology is included to help the reader understand such items as the Riesz theorem, detailing its proofs and statements. This work will help readers apply measure theory to probability theory, guiding them to understand the theorems rather than merely follow directions.

Table of Contents:
Preface Set Theory and General Topology Compactness and Continuous Functions Banach Spaces Hilbert Spaces Calculus in Banach Space Locally Convex Topological Vector Spaces Measures and Measurable Functions The Abstract Lebesgue Integral The Construction of Measures Lebesgue Measure Product Measure The Lp Spaces Representation Theorems Fundamental Theorem of Calculus General Radon Measures Fourier Transforms Probability Weak Derivatives Hausdorff Measures The Area Formula The Coarea Formula Fourier Analysis in Rn Integration for Vector Valued Functions Convex Functions Appendix 1: The Hausdorff Maximal Theorem Appendix 2: Stone's Theorem and Partitions of Unity Appendix 3: Taylor Series and Analytic Functions Appendix 4: The Brouwer Fixed Point Theorem References Index

Review :
"A lucid and effective presentation of...sophisticated material" - Joseph A. Cima, Department of Mathematics, University of North Carolina, Chapel Hill "Ambitious...Conversant...Great" Steven G. Krantz, Department of Math, Washington University, St. Louis, Missouri "The author has chosen his topics well to demonstrate how even the oldest subjects can have a "modern" approach that improves on the original. The text is all business and very readable, especially for the mathematically prepared reader. There is a lot to recommend from the use of this book, not the least of which is the fact that the reader participates in the continuing documentation of "modern" mathematical analysis." -Timothy Hall, PQI Consulting