**Graduate Studies in Mathematics**

Volume: 4;
1994;
395 pp;
Hardcover

MSC: Primary 26; 28;

Print ISBN: 978-0-8218-3805-1

Product Code: GSM/4

List Price: $56.00

Individual Member Price: $44.80

**Electronic ISBN: 978-1-4704-2066-6
Product Code: GSM/4.E**

List Price: $56.00

Individual Member Price: $44.80

# The Integrals of Lebesgue, Denjoy, Perron, and Henstock

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*Russell A. Gordon*

This book provides an elementary, self-contained presentation of the integration processes developed by Lebesgue, Denjoy, Perron, and Henstock. The Lebesgue integral and its essential properties are first developed in detail. The other three integrals are all generalizations of the Lebesgue integral that satisfy the ideal version of the Fundamental Theorem of Calculus: if \(F\) is differentiable on the interval \([a,b]\), then \(F'\) is integrable on \([a,b]\) and \(\int _a^b F'= F(b) - F(a)\). One of the book's unique features is that the Denjoy, Perron, and Henstock integrals are each developed fully and carefully from their corresponding definitions. The last part of the book is devoted to integration processes which satisfy a theorem analogous to the Fundamental Theorem, in which \(F\) is approximately differentiable. This part of this book is preceded by a detailed study of the approximate derivative and ends with some open questions. This book contains over 230 exercises (with solutions) that illustrate and expand the material in the text. It would be an excellent textbook for first-year graduate students who have background in real analysis.

#### Table of Contents

# Table of Contents

## The Integrals of Lebesgue, Denjoy, Perron, and Henstock

- Cover Cover11 free
- Title iii4 free
- Copyright iv5 free
- Contents vii8 free
- Preface ix10 free
- Chapter 1. Lebesgue Measure 114 free
- Chapter 2. Measurable Functions 1730
- Chapter 3. The Lebesgue Integral 2942
- Chapter 4. Bounded Variation and Absolute Continuity 4962
- Chapter 5. Darboux and Baire Class One Functions 6982
- Chapter 6. Functions of Generalized Bounded Variation 89102
- Chapter 7. The Denjoy Integral 107120
- Chapter 8. The Perron Integral 121134
- Chapter 9. The Henstock Integral 137150
- Chapter 10. The McShane Integral 157170
- Chapter 11. Equivalence of Integrals 169182
- Chapter 12. Integration by Parts 181194
- Chapter 13. Convergence Theorems 201214
- Chapter 14. Approximate Derivatives 223236
- Chapter 15. The Khintchine Integral 237250
- Chapter 16. The Approximately Continuous Henstock Integral 245258
- Chapter 17. The Approximately Continuous Perron Integral 259272
- Solutions to Exercises 271284
- References 389402
- Notation Index 391404
- Subject Index 393406
- Back Cover Back Cover1409

#### Readership

First year graduate students in mathematics.