Theories of integration : the integrals of Riemann, Lebesgue, Henstock-Kurzweil, and Mcshane / Douglas S. Kurtz, Charles W. Swartz.

Format:

Book

Author/Creator:

Kurtz, Douglas S.

Contributor:

Kurzweil, Jaroslav.

Sabin W. Colton, Jr., Memorial Fund.

Series:

Series in real analysis ; v. 9.

Series in real analysis ; v. 9

Language:

English

Subjects (All):

Integrals.

Physical Description:

xiii, 268 pages : illustrations ; 24 cm.

Place of Publication:

River Edge, N.J. : World Scientific Pub., 2004.

Summary:

Kurtz and Swartz (both New Mexico State U.) introduce a broad selection of integration theories focusing on the integrals named in the title. They present classical problems in integration theory in historical order to show how new theories were developed to solve problems that earlier ones could not handle. The detail of discussion varies from integral to integral. The four chapters are independent, and each contains 30-60 exercises to make it usable as a text in an introductory real analysis course. Annotation ©2004 Book News, Inc., Portland, OR (booknews.com)

Contents:

1.1 Areas 1

2 Riemann integral 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

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 R[superscript n] 79

3.4 Measurable functions 85

3.5 Lebesgue integral 96

3.6 Riemann and Lebesgue integrals 111

3.7 Mikusinski's characterization of the Lebesgue integral 113

3.8 Fubini's Theorem 117

3.9 The space of Lebesgue integrable functions 122

4 Fundamental Theorem of Calculus and the Henstock-Kurzweil integral 133

4.1 Denjoy and Perron integrals 135

4.2 A General Fundamental Theorem of Calculus 137

4.3 Basic properties 145

4.3.1 Cauchy Criterion 150

4.3.2 The integral as a set function 151

4.4 Unbounded intervals 154

4.5 Henstock's Lemma 162

4.6 Absolute integrability 172

4.6.1 Bounded variation 172

4.6.2 Absolute integrability and indefinite integrals 175

4.6.3 Lattice Properties 178

4.7 Convergence theorems 180

4.8 Henstock-Kurzweil and Lebesgue integrals 189

4.9 Differentiating indefinite integrals 190

4.9.1 Functions with integral 0 195

4.10 Characterizations of indefinite integrals 195

4.10.1 Derivatives of monotone functions 198

4.10.2 Indefinite Lebesgue integrals 203

4.10.3 Indefinite Riemann integrals 204

4.11 The space of Henstock-Kurzweil integrable functions 205

4.12 Henstock-Kurzweil integrals on R[superscript n] 206

5 Absolute integrability and the McShane integral 223

5.3.1 Fundamental Theorem of Calculus 232

5.4 Convergence theorems 234

5.5 The McShane integral as a set function 240

5.6 The space of McShane integrable functions 244

5.7 McShane, Henstock-Kurzweil and Lebesgue integrals 245

5.8 McShane integrals on R[superscript n] 253

5.9 Fubini and Tonelli Theorems 254

5.10 McShane, Henstock-Kurzweil and Lebesgue integrals in R[superscript n] 257.

Notes:

Includes bibliographical references (pages 263-264) and index.

Local Notes:

Acquired for the Penn Libraries with assistance from the Sabin W. Colton, Jr., Memorial Fund.

ISBN:

9812388435

OCLC:

55502425