An Introduction to Lebesgue Integration and Fourier Series

by Wilcox, Howard J.; Myers, David L.

• ISBN13: 9780486682938
• ISBN10: 0486682935
• Format: Paperback
• Copyright: 1995-01-04
• Publisher: Dover Publications

Summary

Undergraduate-level introduction to Riemann integral, measurable sets, measurable functions, Lebesgue integral, other topics. Numerous examples and exercises.

Table of Contents

The Riemann Integral

Definition of the Riemann Integral

Properties of the Riemann Integral

Examples

Drawbacks of the Riemann Integral

Exercises

Measurable Sets

Introduction

Outer Measure

Measurable Sets

Exercises

Properties of Measurable Sets

Countable Additivity

Summary

Borel Sets and the Cantor Set

Necessary and Sufficient Conditions for a Set to be Measurable

Lebesgue Measure for Bounded Sets

Lebesgue Measure for Unbounded Sets

Exercises

Measurable Functions

Definition of Measurable Functions

Preservation of Measurability for Functions

Simple Functions

Exercises

The Lebesgue Integral

The Lebesgue Integral for Bounded Measurable Functions

Simple Functions

Integrability of Bounded Measurable Functions

Elementary Properties of the Integral for Bounded Functions

The Lebesgue Integral for Unbounded Functions

Exercises

Convergence and The Lebesgue Integral

Examples

Convergence Theorems

A Necessary and Sufficient Condition for Riemann Integrability

Egoroff's and Lusin's Theorems and an Alternative Proof of the Lebesgue Dominated Convergence Theorem

Exercises

Function Spaces and £ superscript 2

Linear Spaces

The Space £ superscript 2

Exercises

The £ superscript 2 Theory of Fourier Series

Definition and Examples

Elementary Properties

£ superscript 2 Convergence of Fourier Series

Exercises

Pointwise Convergence of Fourier Series

An Application: Vibrating Strings

Some Bad Examples and Good Theorems

More Convergence Theorems

Exercises

Appendix

Logic and Sets

Open and Closed Sets

Bounded Sets of Real Numbers

Countable and Uncountable Sets (and discussion of the Axiom of Choice)

Real Functions

Real Sequences

Sequences of Functions

Bibliography

Index

Table of Contents provided by Publisher. All Rights Reserved.