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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. |
The New copy of this book will include any supplemental materials advertised. Please check the title of the book to determine if it should include any access cards, study guides, lab manuals, CDs, etc.
The Used, Rental and eBook copies of this book are not guaranteed to include any supplemental materials. Typically, only the book itself is included. This is true even if the title states it includes any access cards, study guides, lab manuals, CDs, etc.