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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

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

Library of Congress subject headings for this publication: Lebesgue integral, Fourier series