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Introduction to Gauge Integrals

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This book presents the Henstock/Kurzweil integral and the McShane integral. These two integrals are obtained by changing slightly the definition of the Riemann integral. These variations lead to integrals which are much more powerful than the Riemann integral. The Henstock/Kurzweil integral is an unconditional integral for which the fundamental theorem of calculus holds in full generality, while the McShane integral is equivalent to the Lebesgue integral in Euclidean spaces.

A basic knowledge of introductory real analysis is required of the reader, who should be familiar with the fundamental properties of the real numbers, convergence, series, differentiation, continuity, etc.

Author(s): Charles Swartz
Publisher: World Scientific
Year: 2001

Language: English
Pages: 169
City: Singapore; River Edge, NJ

Preface......Page 6
Contents......Page 10
1 Introduction to the Gauge or Henstock-Kurzweil Integral......Page 12
2 Basic Properties of the Gauge Integral......Page 24
3 Henstock's Lemma and Improper Integrals......Page 34
4 The Gauge Integral over Unbounded Intervals......Page 44
5 Convergence Theorems......Page 60
6 Integration over More General Sets: Lebesgue Measure......Page 74
7 The Space of Gauge Integrable Functions......Page 84
8 Multiple Integrals and Fubini's Theorem......Page 92
9.1 Definition and Basic Properties......Page 110
9.2 Convergence Theorems......Page 116
9.3 Integrability of Products and Integration by Parts......Page 121
9.4 More General Convergence Theorems......Page 123
9.5 The Space of McShane Integrable Functions......Page 126
9.6 Multiple Integrals and Fubini's Theorem......Page 127
10 McShane Integrability is Equivalent to Absolute Henstock-Kurzweil Integrability......Page 132
Appendix 1: The Riemann Integral......Page 138
Appendix 2: Functions of Bounded Variations......Page 140
Appendix 3: Differentiating Indefinite Integrals......Page 146
Appendix 4: Equivalence of Lebesgue and McShane Integrals......Page 148
Appendix 5: Change of Variable in Multiple Integrals......Page 152
Bibliography......Page 160
Index......Page 166