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This title in other editions

An Introduction to Lebesgue Integration and Fourier Series (Dover Books on Advanced Mathematics)

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An Introduction to Lebesgue Integration and Fourier Series (Dover Books on Advanced Mathematics) Cover

 

Synopses & Reviews

Publisher Comments:

This book arose out of the authors' desire to present Lebesgue integration and Fourier series on an undergraduate level, since most undergraduate texts do not cover this material or do so in a cursory way. The result is a clear, concise, well-organized introduction to such topics as the Riemann integral, measurable sets, properties of measurable sets, measurable functions, the Lebesgue integral, convergence and the Lebesgue integral, pointwise convergence of Fourier series and other subjects.

The authors not only cover these topics in a useful and thorough way, they have taken pains to motivate the student by keeping the goals of the theory always in sight, justifying each step of the development in terms of those goals. In addition, whenever possible, new concepts are related to concepts already in the student's repertoire.

Finally, to enable readers to test their grasp of the material, the text is supplemented by numerous examples and exercises. Mathematics students as well as students of engineering and science will find here a superb treatment, carefully thought out and well presented , that is ideal for a one semester course. The only prerequisite is a basic knowledge of advanced calculus, including the notions of compactness, continuity, uniform convergence and Riemann integration.

Synopsis:

Clear and concise introductory treatment for undergraduates covers Riemann integral, measurable sets and their properties, measurable functions, Lebesgue integral and convergence, pointwise conversion of Fourier series, other subjects. 1978 edition.

Synopsis:

This clear and concise introductory treatment for undergraduates covers the Riemann integral, measurable sets and their properties, measurable functions, the Lebesgue integral and convergence, pointwise conversion of the Fourier series, and other subjects. Numerous examples and exercises supplement the text. Basic knowledge of advanced calculus is the sole prerequisite. 1978 edition.
 

Synopsis:

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

Table of Contents

Chapter 1. The Riemann Integral

  1. Definition of the Riemann Integral

  2. Properties of the Riemann Integral

  3. Examples

  4. Drawbacks of the Riemann Integral

  5. Exercises

Chapter 2. Measurable Sets

  6. Introduction

  7. Outer Measure

  8. Measurable Sets

  9. Exercises

Chapter 3. Properties of Measurable Sets

  10. Countable Additivity

  11. Summary

  12. Borel Sets and the Cantor Set

  13. Necessary and Sufficient Conditions for a Set to be Measurable

  14. Lebesgue Measure for Bounded Sets

  15. Lebesgue Measure for Unbounded Sets

  16. Exercises

Chapter 4. Measurable Functions

  17. Definition of Measurable Functions

  18. Preservation of Measurability for Functions

  19. Simple Functions

  20. Exercises

Chapter 5. The Lebesgue Integral

  21. The Lebesgue Integral for Bounded Measurable Functions

  22. Simple Functions

  23. Integrability of Bounded Measurable Functions

  24. Elementary Properties of the Integral for Bounded Functions

  25. The Lebesgue Integral for Unbounded Functions

  26. Exercises

Chapter 6. Convergence and The Lebesgue Integral

  27. Examples

  28. Convergence Theorems

  29. A Necessary and Sufficient Condition for Riemann Integrability

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

  31. Exercises

Chapter 7. Function Spaces and £ superscript 2

  32. Linear Spaces

  33. The Space £ superscript 2

  34. Exercises

Chapter 8. The £ superscript 2 Theory of Fourier Series

  35. Definition and Examples

  36. Elementary Properties

  37. £ superscript 2 Convergence of Fourier Series

  38. Exercises

Chapter 9. Pointwise Convergence of Fourier Series

  39. An Application: Vibrating Strings

  40. Some Bad Examples and Good Theorems

  41. More Convergence Theorems

  42. 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

Product Details

ISBN:
9780486682938
Author:
Wilcox, Howard J.
Author:
Myers, David L.
Author:
Mathematics
Publisher:
Dover Publications
Location:
New York :
Subject:
Calculus
Subject:
Integrals, generalized
Subject:
Fourier series
Subject:
Lebesgue integral
Subject:
General Mathematics
Subject:
Advanced
Subject:
Mathematics-Calculus
Edition Number:
Dover ed.
Edition Description:
Trade Paper
Series:
Dover Books on Mathematics
Publication Date:
19950131
Binding:
TRADE PAPER
Language:
English
Illustrations:
Yes
Pages:
192
Dimensions:
9.25 x 6.13 in 0.61 lb

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Related Subjects

Reference » Science Reference » General
Science and Mathematics » Chemistry » General
Science and Mathematics » Mathematics » Calculus » General
Science and Mathematics » Mathematics » Functional Analysis
Science and Mathematics » Mathematics » General
Science and Mathematics » Physics » Solid State Physics

An Introduction to Lebesgue Integration and Fourier Series (Dover Books on Advanced Mathematics) New Trade Paper
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Product details 192 pages Dover Publications - English 9780486682938 Reviews:
"Synopsis" by , Clear and concise introductory treatment for undergraduates covers Riemann integral, measurable sets and their properties, measurable functions, Lebesgue integral and convergence, pointwise conversion of Fourier series, other subjects. 1978 edition.
"Synopsis" by ,
This clear and concise introductory treatment for undergraduates covers the Riemann integral, measurable sets and their properties, measurable functions, the Lebesgue integral and convergence, pointwise conversion of the Fourier series, and other subjects. Numerous examples and exercises supplement the text. Basic knowledge of advanced calculus is the sole prerequisite. 1978 edition.
 
"Synopsis" by ,
Undergraduate-level introduction to Riemann integral, measurable sets, measurable functions, Lebesgue integral, other topics. Numerous examples and exercises.
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