Integral: An Easy Approach after Kurzweil and

Integration has a long history: its roots can be traced as far back as the ancient Greeks. The first genuinely rigorous definition of an integral was that given by Riemann, and further (more general, and so more useful) definitions have since been given by Lebesgue, Denjoy, Perron, Kurzweil and Henstock, and this culminated in the work of McShane. This textbook provides an introduction to this theory, and it presents a unified yet elementary approach that is suitable for beginning graduate and final year undergraduate students.

Preface

viii

List of Symbols

xi

Introduction

1

(21)

Historical remarks

1

(2)

Notation and the Riemann definition

3

(3)

Basic theorems, upper and lower integrals

6

(4)

Differentiability, continuity and integrability

10

(6)

Limit and R-integration

16

(2)

Exercises

18

(4)

Basic Theory

22

(54)

Introduction

22

(1)

Motivation

22

(3)

Cousin's lemma

25

(4)

Applications of Cousin's lemma

26

(3)

The definition

29

(3)

Basic theorems

32

(14)

The Fundamental Theorem of calculus

46

(4)

Consequences of the Fundamental Theorem

50

(5)

Improper integrals

55

(4)

Integrals over unbounded intervals

59

(5)

Alternative approach to integration over unbounded intervals

64

(2)

Negligible sets

66

(3)

Complex valued function

69

(2)

Exercises

71

(5)

Development of the Theory

76

(75)

Equivalent forms of the definition

76

(5)

Henstock's lemma

81

(2)

Functions of bounded variation

83

(3)

Absolute integrability

86

(2)

Limit and KH-integration

88

(12)

Absolute continuity

100

(4)

Equiintegrability

104

(6)

The second mean value theorem

108

(2)

Differentiation of integrals

110

(2)

Characterization of the KH-integral

112

(3)

Lebesgue points, approximation by step functions

115

(2)

Measurable functions and sets

117

(10)

A non-measurable set

126

(1)

The McShane integral

127

(8)

A short proof

135

(1)

The Lebesgue integral

135

(5)

F. Riesz' definition

139

(1)

Quick proofs

140

(1)

Differentiation almost everywhere

140

(5)

Exercises

145

(6)

The SL-integral

151

(24)

The strong Luzin condition

151

(4)

SL-integration

155

(7)

Limit and SL-integration

162

(7)

Equivalence with the KH-integral

169

(2)

Exercises

171

(4)

Generalized AC Functions

175

(27)

Prologue

175

(2)

Uniformly AC functions

177

(2)

AC* and VB* on a set

179

(8)

ACG* functions

187

(3)

Controlled convergence

190

(7)

Exercise

197

(5)

Integration in Several Dimensions

202

(50)

Introduction

202

(1)

Sets in Rn

203

(1)

Divisions, partitions

204

(6)

The definition

210

(5)

Basic theorems

215

(5)

Prelude to Fubini's theorem

217

(3)

Other theorems in Rn

220

(9)

Negligible sets

220

(1)

Henstock's lemma

220

(1)

Absolute integrability

221

(2)

Convergence, measurability, AC

223

(6)

The Fubini theorem

229

(7)

Change of variables

236

(12)

Introductory examples

236

(4)

Notation, lemmas

240

(1)

The theorem

241

(7)

Exercises

248

(4)

Some Applications

252

(47)

Introduction

252

(1)

A line integral

253

(26)

Green's theorem

267

(7)

The Cauchy theorem

274

(5)

Differentiation of series

279

(3)

Dirichlet's problem and the Poisson integral

282

(4)

Summability of Fourier series

286

(3)

Fourier series and the space L2

289

(6)

Exercises

295

(4)

Appendix 1 Supplements

299

(6)

Bibliography

305

(3)

Index

308

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Integral: An Easy Approach after Kurzweil and Henstock

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Table of Contents

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Integral: An Easy Approach after Kurzweil and Henstock

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Table of Contents

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