Theories of Integration: The Integrals of Riemann, Lebesgue, Henstock-kurzweil, and Mcshane, Second Edition

by Kurtz, Douglas S.; Charles W Swartz

• ISBN13: 9789814368995
• ISBN10: 9814368997
• Format: Hardcover
• Copyright: 2011-10-31
• Publisher: World Scientific Pub Co Inc

Summary

The book uses classical problems to motivate a historical development of the integration theories of Riemann, Lebesgue, HenstockKurzweil 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.

Table of Contents

Preface to the First Edition	p. vii
Preface to the Second Edition	p. xi
Introduction	p. 1
Areas	p. 1
Exercises	p. 9
Riemann integral	p. 11
Riemann's definition	p. 11
Basic properties	p. 15
Cauchy criterion	p. 18
Darboux's definition	p. 20
Necessary and sufficient conditions for Darboux integrability	p. 24
Equivalence of the Riemann and Darboux definitions	p. 25
Lattice properties	p. 27
Integrable functions	p. 30
Additivity of the integral over intervals	p. 31
Fundamental Theorem of Calculus	p. 33
Integration by parts and substitution	p. 37
Characterizations of integrability	p. 38
Lebesgue measure zero	p. 41
Improper integrals	p. 42
Exercises	p. 46
Convergence theorems and the Lebesgue integral	p. 53
Lebesgue's descriptive definition of the integral	p. 56
Measure	p. 60
Outer measure	p. 60
Lebesgue measure	p. 64
The Cantor set	p. 78
Lebesgue measure in Rn	p. 79
Measurable functions	p. 85
Lebesgue integral	p. 96
Integrals depending on a parameter	p. 111
Riemann and Lebesgue integrals	p. 113
Mikusinski's characterization of the Lebesgue integral	p. 114
Fubini's Theorem	p. 119
Convolution	p. 123
The space of Lebesgue integrable functions	p. 129
Exercises	p. 139
Fundamental Theorem of Calculus and the Henstock-Kurzweil integral	p. 147
Denjoy and Perron integrals	p. 149
A General Fundamental Theorem of Calculus	p. 151
Basic properties	p. 159
Cauchy criterion	p. 166
The integral as a set function	p. 167
Unbounded intervals	p. 171
Henstock's Lemma	p. 178
Absolute integrability	p. 188
Bounded variation	p. 188
Absolute integrability and indefinite integrals	p. 192
Lattice properties	p. 194
Convergence theorems	p. 196
Henstock-Kurzweil and Lebesgue integrals	p. 210
Differentiating indefinite integrals	p. 212
Functions with integral 0	p. 217
Characterizations of indefinite integrals	p. 217
Derivatives of monotone functions	p. 220
Indefinite Lebesgue integrals	p. 224
Indefinite Riemann integrals	p. 226
The space of Henstock-Kurzweil integrable functions	p. 227
Henstock-Kurzweil integrals on Rn	p. 231
Exercises	p. 238
. Absolute integrability and the McShane integral	p. 247
Definitions	p. 248
Basic properties	p. 251
Absolute integrability	p. 253
Fundamental Theorem of Calculus	p. 256
Convergence theorems	p. 259
The McShane integral as a set function	p. 266
The space of McShane integrable functions	p. 270
McShane, Henstock-Kurzweil and Lebesgue integrals	p. 270
McShane integrals on Rn	p. 279
Fubini and Tonelli Theorems	p. 280
McShane, Henstock-Kurzweil and Lebesgue integrals in Rn	p. 283
Exercises	p. 284
Bibliography	p. 289
Index	p. 291
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