Integral: An Easy Approach after Kurzweil and Henstock

by Lee Peng Yee , Rudolf Vyborny

• ISBN13: 9780521779685
• ISBN10: 0521779685
• Format: Paperback
• Copyright: 2000-04-28
• Publisher: Cambridge University Press

Summary

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.

Table of Contents

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