Measure, Integration and Functional Analysis

✍ Scribed by Robert B. Ash

• Publisher: Academic Press
• Year: 1972
• Tongue: English
• Leaves: 299
• Category: Library

✦ Synopsis

Measure, Integration, and Functional Analysis deals with the mathematical concepts of measure, integration, and functional analysis. The fundamentals of measure and integration theory are discussed, along with the interplay between measure theory and topology. Comprised of four chapters, this book begins with an overview of the basic concepts of the theory of measure and integration as a prelude to the study of probability, harmonic analysis, linear space theory, and other areas of mathematics. The reader is then introduced to a variety of applications of the basic integration theory developed in the previous chapter, with particular reference to the Radon-Nikodym theorem. The third chapter is devoted to functional analysis, with emphasis on various structures that can be defined on vector spaces. The final chapter considers the connection between measure theory and topology and looks at a result that is a companion to the monotone class theorem, together with the Daniell integral and measures on topological spaces. The book concludes with an assessment of measures on uncountably infinite product spaces and the weak convergence of measures. This book is intended for mathematics majors, most likely seniors or beginning graduate students, and students of engineering and physics who use measure theory or functional analysis in their work.

✦ Table of Contents

Preface

Summary of Notation
1 Sets
2 Real Numbers
3 Functions
4 Topology
5 Vector Spaces
6 Zorn's Lemma

1 Fundamentals of Measure and Integration Theory
1.1 Introduction
1.2 Fields, ó-Fields, and Measures
1.3 Extension of Measures
1.4 Lebesgue-Stieltjes Measures and Distribution Functions
1.5 Measurable Functions and Integration
1.6 Basic Integration Theorems
1.7 Comparison of Lebesgue and Riemann Integrals

2 Further Results in Measure and Integration Theory
2.1 Introduction
2.2 Radon-Nikodym Theorem and Related Results
2.3 Applications to Real Analysis
2.4 L^p Spaces
2.5 Convergence of Sequences of Measurable Functions
2.6 Product Measures and Fubini's Theorem
2.7 Measures on Infinite Product Spaces
2.8 References

3 Introduction to Functional Analysis
3.1 Introduction
3.2 Basic Properties of Hilbert Spaces
3.3 Linear Operators on Normed Linear Spaces
3.4 Basic Theorems of Functional Analysis
3.5 Some Properties of Topological Vector Spaces
3.6 References

4 The Interplay between Measure Theory and Topology
4.1 Introduction
4.2 The Daniell Integral
4.4 Measures on Uncountably Infinite Product Spaces
4.5 Weak Convergence of Measures
4.6 References

Appendix on General Topology
A1 Introduction
A2 Convergence
A3 Product and Quotient Topologies
A4 Separation Properties and Other Ways of Classifying Topological Spaces
A5 Compactness
A6 Semicontinuous Functions
A7 The Stone-Weierstrass Theorem
A8 Topologies on Function Spaces
A9 Complete Metric Spaces and Category Theorems
A10 Uniform Spaces

Bibliography

Solutions to Problems
Chapter 1
Chapter 2
Chapter 3
Chapter 4

Subject Index