Theories Of Integration: The Integrals Of Riemann, Lebesgue, Henstock-kurzweil, And Mcshane (Second Edition)

by Charles W Swartz, Douglas S Kurtz

Overview

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.

Product Details

ISBN-13:	9789814368995
Publisher:	World Scientific Publishing Company, Incorporated
Publication date:	11/02/2011
Series:	Series In Real Analysis , #13
Pages:	312
Product dimensions:	6.10(w) x 9.00(h) x 1.00(d)

Table of Contents

Preface to the First Edition vii

Preface to the Second Edition xi

1 Introduction 1

1.1 Areas 1

1.2 Exercises 9

2 Riemann integral 11

2.1 Riemann's definition 11

2.2 Basic properties 15

2.3 Cauchy criterion 18

2.4 Darboux's definition 20

2.4.1 Necessary and sufficient conditions for Darboux integrability 24

2.4.2 Equivalence of the Riemann and Darboux definitions 25

2.4.3 Lattice properties 27

2.4.4 Integrable functions 30

2.4.5 Additivity of the integral over intervals 31

2.5 Fundamental Theorem of Calculus 33

2.5.1 Integration by parts and substitution 37

2.6 Characterizations of integrability 38

2.6.1 Lebesgue measure zero 41

2.7 Improper integrals 42

2.8 Exercises 46

3 Convergence theorems and the Lebesgue integral 53

3.1 Lebesgue's descriptive definition of the integral 56

3.2 Measure 60

3.2.1 Outer measure 60

3.2.2 Lebesgue measure 64

3.2.3 The Cantor set 78

3.3 Lebesgue measure in Rⁿ 79

3.4 Measurable functions 85

3.5 Lebesgue integral 96

3.5.1 Integrals depending on a parameter 111

3.6 Riemann and Lebesgue integrals 113

3.7 Mikusinski's characterization of the Lebesgue integral 114

3.8 Fubini's Theorem 119

3.8.1 Convolution 123

3.9 The space of Lebesgue integrable functions 129

3.10 Exercises 139

4 Fundamental Theorem of Calculus and the Henstock-Kurzweil integral 147

4.1 Denjoy and Perron integrals 149

4.2 A General Fundamental Theorem of Calculus 151

4.3 Basic properties 159

4.3.1 Cauchy criterion 166

4.3.2 The integral as a set function 167

4.4 Unbounded intervals 171

4.5 Henstock's Lemma 178

4.6 Absolute integrability 188

4.6.1 Bounded variation 188

4.6.2 Absolute integrability and indefinite integrals 192

4.6.3 Lattice properties 194

4.7 Convergence theorems 196

4.8 Henstock-Kurzweil and Lebesgue integrals 210

4.9 Differentiating indefinite integrals 212

4.9.1 Functions with integral 0 217

4.10 Characterizations of indefinite integrals 217

4.10.1 Derivatives of monotone functions 220

4.10.2 Indefinite Lebesgue integrals 224

4.10.3 Indefinite Riemann integrals 226

4.11 The space of Henstock-Kurzweil integrable functions 227

4.12 Henstock-Kurzweil integrals on Rⁿ 231

4.13 Exercises 238

5 . Absolute integrability and the McShane integral 247

5.1 Definitions 248

5.2 Basic properties 251

5.3 Absolute integrability 253

5.3.1 Fundamental Theorem of Calculus 256

5.4 Convergence theorems 259

5.5 The McShane integral as a set function 266

5.6 The space of McShane integrable functions 270

5.7 McShane, Henstock-Kurzweil and Lebesgue integrals 270

5.8 McShane integrals on Rⁿ 279

5.9 Fubini and Tonelli Theorems 280

5.10 McShane, Henstock-Kurzweil and Lebesgue integrals in Rⁿ 283

5.11 Exercises 284

Bibliography 289

Index 291