A chapter on the Bochner integral - used widely in an applications setting by engineers and others, but its actual theorems are often difficult to find in the literatureAn accessible text for graduate students who have not had extensive preparation in analysisInformation on introductory topology helping students in understanding topics like the Riesz representation theoryVarious proofs on partial differential equations not usually found in the literatureExercises that extend the theorems and supply illustrative examples
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
Set Theory and General Topology
Compactness and Continuous Functions
Calculus in Banach Space
Locally Convex Topological Vector Spaces
Measures and Measurable Functions
The Abstract Lebesgue Integral
The Construction of Measures
The Lp Spaces
Fundamental Theorem of Calculus
General Radon Measures
The Area Formula
The Coarea Formula
Fourier Analysis in Rn
Integration for Vector Valued 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
"A lucid and effective presentation of...sophisticated material"
- Joseph A. Cima, Department of Mathematics, University of North Carolina, Chapel Hill
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