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.
Author(s): Douglas S Kurtz, Charles W Swartz
Series: Series in Real Analysis: Volume 13
Edition: 2
Year: 2011
Language: English
Pages: 312
Contents
Preface to the First Edition
Preface to the Second Edition
1. Introduction
1.1 Areas
1.2 Exercises
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
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 Rn
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
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
4.9.1 Functions with integral 0
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 Rn
4.13 Exercises
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 Rn
5.9 Fubini and Tonelli Theorems
5.10McShane, Henstock-Kurzweil and Lebesgue integrals in Rn
5.11 Exercises
Bibliography
Index
