An Introduction to Lebesgue Integration and Fourier Series

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ISBN: 9780486682938 | 0486682935
Cover: Paperback
Copyright: 1/4/1995

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Summary

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

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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.

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An Introduction to Lebesgue Integration and Fourier Series

Summary

Table of Contents

Supplemental Materials

FREE SHIPPING

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An Introduction to Lebesgue Integration and Fourier Series

Summary

Table of Contents

Supplemental Materials

Digital License