Measure and Integration : A Concise Introduction to Real Analysis

by Leonard F. Richardson (Louisiana State Univ. )

• ISBN13: 9780470501146
• ISBN10: 0470501146
• Format: eBook
• Copyright: 2009-07-01
• Publisher: Wiley

Summary

A uniquely accessible book for general measure and integration, emphasizing the real line, Euclidean space, and the underlying role of translation in real analysis Measure and Integration: A Concise Introduction to Real Analysis presents the basic concepts and methods that are important for successfully reading and understanding proofs. Blending coverage of both fundamental and specialized topics, this book serves as a practical and thorough introduction to measure and integration, while also facilitating a basic understanding of real analysis. The author develops the theory of measure and integration on abstract measure spaces with an emphasis of the real line and Euclidean space. Additional topical coverage includes: Measure spaces, outer measures, and extension theorems Lebesgue measure on the line and in Euclidean space Measurable functions, Egoroff's theorem, and Lusin's theorem Convergence theorems for integrals Product measures and Fubini's theorem Differentiation theorems for functions of real variables Decomposition theorems for signed measures Absolute continuity and the Radon-Nikodym theorem Lp spaces, continuous-function spaces, and duality theorems Translation-invariant subspaces of L2 and applications The book's presentation lays the foundation for further study of functional analysis, harmonic analysis, and probability, and its treatment of real analysis highlights the fundamental role of translations. Each theorem is accompanied by opportunities to employ the concept, as numerous exercises explore applications including convolutions, Fourier transforms, and differentiation across the integral sign. Providing an efficient and readable treatment of this classical subject, Measure and Integration: A Concise Introduction to Real Analysis is a useful book for courses in real analysis at the graduate level. It is also a valuable reference for practitioners in the mathematical sciences.

Table of Contents

Preface

Acknowledgments

Introduction

History of the Subject

History of the Idea 1

Deficiencies of the Riemann Integral

Exercises

Motivation for the Lebesgue Integral

Fields, Borel Fields and Measures

Fields, Monotone Classes, and Borel Fields

Exercises

Exercises

Exercises

Exercises

Additive Measures

Exercises

Carathéodory Outer Measure

Exercises

E. Hopf's Extension Theorem

Exercises

Fields, _Fields, and Measures Inherited by a Subset

Lebesgue Measure

The Finite Interval [-N,N)

Measurable Sets, Borel Sets, and the Real Line

Exercises

Lebesgue Measure on R

Exercises

Measure Spaces and Completions

Minimal Completion of a Measure Space

A Nonmeasurable Set

Semimetric Space of Measurable Sets

Exercises

Exercises

Lebesgue Measure in Rn

Exercises

Jordan Measure in Rn

Exercises

Exercises

Measurable Functions

Measurable Functions

Exercises 58

Baire Functions of Measurable Functions

Limits of Measurable Functions

Exercises

Simple Functions and Egoroff's Theorem

Exercises

Double Sequences

Convergence in Measure

Exercises

Lusin's Theorem

Exercises

The Integral

Special Simple Functions

Exercises

Exercises

Extending the Domain of the Integral

The Class L+ of nonnegative Measurable Functions

The Class L of Lebesgue Integrable Functions

Exercises

Convex Functions and Jensen?s Inequality

Exercises

Lebesgue Dominated Convergence Theorem

Exercises

Monotone Convergence and Fatou's Theorem

Exercises

Exercises

Completeness of L1(X,A, ?) and the Pointwise Convergence Lemma

Exercises

Exercises

Complex Valued Functions

Exercises

Product Measures and Fubini's Theorem

Product Measures

Fubini's Theorem

Exercises

Comparison of Lebesgue and Riemann Integrals

Exercises

Functions of a Real Variable

Functions of Bounded Variation

Exercises

A Fundamental Theorem for the Lebesgue Integral

Lebesgue's Theorem and Vitali's Covering Theorem

Absolutely Continuous & Singular Functions

Exercises

Exercises

Exercises

General Countably Additive Set Functions

Hahn Decomposition Theorem

Exercises

RadonNikodym Theorem

Exercises 163

Lebesgue Decomposition Theorem 165

Exercises

Examples of Dual Spaces from Measure Theory

The Banach Space Lp(X,A, ¿)

Exercises

The Dual of a Banach Space

Exercises

Exercises

The Dual Space of Lp(X,A, ¿)

Exercises

Hilbert Space, its Dual, and L2(X,A, ¿)

Exercises

RieszMarkovSaksKakutani Theorem

Exercises

Translation Invariance in Real Analysis

An Orthonormal Basis for L2(T)

Exercises

Closed Invariant Subspaces of L2(T)

Integration of Hilbert Space Valued Functions

Spectrum of a Subset of L2(T)

Exercises

Schwartz Functions: Fourier Transform and Inversion

Exercises

Closed Invariant Subspaces of L2(R)

The Fourier Transform in L2(R)

Translation Invariant Subspaces of L2(R)

The Fourier Transform and Direct Integrals

Exercises

Irreducibility of L2(R) Under Translations and Rotations

Position and Momentum Operators

The Heisenberg Group

Exercises

The BanachTarski Theorem

The Limits to Countable Additivity

References

Table of Contents provided by Publisher. All Rights Reserved.