The Integrals of Lebesgue, Denjoy, Perron, and Henstock (Graduate Studies in Mathematics, 4)

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 ∫baF′=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. Readership: First year graduate students in mathematics. Cover Graduate Studies in Mathematics VOLUME 4 The Integrals of Lebesgue, Denjoy, Perron, and Henstock Copyright © Copyright 1994 by the American Mathematical Society ISBN 0-8218-3805-9 QA312.G63 1994 515' .42--dc20 LCCN 94-19080 Dedicated To Brenda and Charles Contents Preface Chapter 1. Lebesgue Measure Exercises Chapter 2. Measurable Functions Exercises Chapter 3. The Lebesgue Integral Exercises Chapter 4. Bounded Variation and Absolute Continuity Exercises Chapter 5. Darboux and Baire Class One Functions Exercises Chapter 6. Functions of Generalized Bounded Variation Exercises Chapter 7. The Denjoy Integral Exercises Chapter 8. The Perron Integral Exercises Chapter 9. The Henstock Integral Exercises Chapter 10. The McShane Integral Exercises Chapter 11. Equivalence of Integrals Exercises Chapter 12. Integration by Parts Exercises Chapter 13. Convergence Theorems Exercises Chapter 14. Approximate Derivatives Exercises Chapter 15. The Khintchine Integral Exercises Chapter 16. The Approximately Continuous Henstock Integral Exercises Chapter 17. The Approximately Continuous Perron Integral Exercises Solutions to Exercises Chapter 1 Exercises. Chapter 2 Exercises Chapter 3 Exercises Chapter 4 Exercises. Chapter 5 Exercises. Chapter 6 Exercises. Chapter 7 Exercises Chapter 8 Exercises. Chapter 9 Exercises Chapter 10 Exercises Chapter 11 Exercises. Chapter 12 Exercises Chapter 13 Exercises Chapter 14 Exercises Chapter 15 Exercises Chapter 16 Exercises Chapter 17 Exercises. References Notation Index Subject Index Back Cover 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 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. 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 for differentiable functions on an interval. 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 for approximately differentiable functions. This part of the 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. This is an excellent textbook for graduate students who have backgrounds in real analysis. Chapter 1. Lebesgue Measure Chapter 2. Measurable Functions Chapter 3. The Lebesgue Integral Chapter 4. Bounded Variation And Absolute Continuity Chapter 5. Darboux And Baire Class One Functions Chapter 6. Functions Of Generalized Bounded Variation Chapter 7. The Denjoy Integral Chapter 8. The Perron Integral Chapter 9. The Henstock Integral Chapter 10. The Mcshane Integral Chapter 11. Equivalence Of Integrals Chapter 12. Integration By Parts Chapter 13. Convergence Theorems Chapter 14. Approximate Derivatives Chapter 15. The Khintchine Integral Chapter 16. The Approximately Continuous Henstock Integral Chapter 17. The Approximately Continuous Perron Integral Solutions To Exercises Russell A. Gordon. Includes Bibliographical References And Index. Offers an elementary, self-contained presentation of the integration processes developed by Lebesgue, Denjoy, Perron, and Henstock. This book contains over 230 exercises (with solutions) that illustrate and expand the material. It is suitable for first-year graduate students who have background in real analysis.