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