The book uses classical problems to motivate a historical development of the integration theories of Riemann, Lebesgue, Henstock–Kurzweil and McShane, showing how new theories of integration were developed to solve problems that earlier integration theories could not handle. It develops the basic properties of each integral in detail and provides comparisons of the different integrals. The chapters covering each integral are essentially independent and could be used separately in teaching a portion of an introductory real analysis course. There is a sufficient supply of exercises to make this book useful as a textbook.
Readership: Upper-level undergraduate students, beginning graduate students, lecturers and researchers interested in integration theory.
Author(s): Douglas S Kurtz, Charles W Swartz
Series: Series in Real Analysis, Vol. 13
Edition: Second Edition
Publisher: World Scientific Publishing Company
Year: 2011
Language: English
Commentary: Don't Change anything, Please!
Pages: C, XV, 294, B
Tags: Математика;Математический анализ;Дифференциальное и интегральное исчисление;
Cover
Series in Real Analysis - Vol. 13
Published Books in this Series
THEORIES OF INTEGRATION: The Integrals of Riemann, Lebesgue, Henstock-Kurzweil, and McShane, Second Edition
Copyright© 2012 by World Scientific Publishing
ISBN-13 978-981-4368-99-5
ISBN-10 981-4368-99-7
Dedication
Preface to the First Edition
Preface to the Second Edition
Contents
Chapter 1 Introduction
1.1 Areas
1. 2 Exercises
Chapter 2 Riemann integral
2.1 Riemann's definition
2.2 Basic properties
2.3 Cauchy criterion
2.4 Darboux's definition
2.4.1 Necessary and sufficient conditions for Darboux integrability
2.4.2 Equivalence of the Riemann and Darboux definitions
2.4.3 Lattice properties
2.4.4 Integrable functions
2.4.5 Additivity of the integral over intervals
2.5 Fundamental Theorem of Calculus
2.5.1 Integration by parts and substitution
2.6 Characterizations of integrability
2.6.1 Lebesgue measure zero
2. 7 Improper integrals
2.8 Exercises
Chapter 3 Convergence theorems and the Lebesgue integral
3.1 Lebesgue's descriptive definition of the integral
3.2 Measure
3.2.1 Outer measure
3.2.2 Lebesgue measure
3.2.3 The Cantor set
3.3 Lebesgue measure in R^n
3.4 Measurable functions
3.5 Lebesgue integral
3.5.1 Integrals depending on a parameter
3.6 Riemann and Lebesgue integrals
3.7 Mikusinski's characterization of the Lebesgue integral
3.8 Fubini's Theorem
3.8.1 Convolution
3.9 The space of Lebesgue integrable functions
3.10 Exercises
Chapter 4 Fundamental Theorem of Calculus and the Henstock-Kurzweil integral
4.1 Denjoy and Perron integrals
4.2 A General Fundamental Theorem of Calculus
4.3 Basic properties
4.3.1 Cauchy criterion
4.3.2 The integral as a set function
4.4 Unbounded intervals
4.5 Henstock's Lemma
4.6 Absolute integrability
4.6.1 Bounded variation
4.6.2 Absolute integrability and indefinite integrals
4.6.3 Lattice properties
4.7 Convergence theorems
4.8 Henstock-Kurzweil and Lebesgue integrals
4.9 Differentiating indefinite integrals
4.10 Characterizations of indefinite integrals
4.10.1 Derivatives of monotone functions
4.10.2 Indefinite Lebesgue integrals
4.10.3 Indefinite Riemann integrals
4.11 The space of Henstock-Kurzweil integrable functions
4.12 Henstock-Kurzweil integrals on R^n
4.13 Exercises
Chapter 5 Absolute integrability and the McShane integral
5.1 Definitions
5.2 Basic properties
5.3 Absolute integrability
5.3.1 Fundamental Theorem of Calculus
5.4 Convergence theorems
5.5 The McShane integral as a set function
5.6 The space of McShane integrable functions
5.7 McShane, Henstock-Kurzweil and Lebesgue integrals
5.8 McShane integrals on R^n
5.9 Fubini and Tonelli Theorems
5.10 McShane, Henstock-Kurzweil and Lebesgue integrals in R^n
5 .11 Exercises
Bibliography
Index
Back Cover
NOTICE: White pages will not be counted at the End of the Front & Back Matter.
