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.
Author(s): Russell A. Gordon
Series: Graduate Studies in Mathematics, Vol.4
Publisher: American Mathematical Society
Year: 1994
Language: English
Pages: C+xi+395+B
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
