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

Howard J. Wilcox, David L. Myers

ISBN 10: 0486682935 / ISBN 13: 9780486682938

Editore: Dover Publications, 1995

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Titolo: An Introduction to Lebesgue Integration and ...

Casa editrice: Dover Publications

Data di pubblicazione: 1995

Legatura: Paperback

Condizione: Good

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Riassunto:

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.

Contenuti:

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