Written by one of the subject's foremost experts, this is the first book on division space integration theory. It is intended to present a unified account of many classes of integrals including the Lebesgue-Bochner, Denjoy-Perron gauge, Denjoy-Hincin, Cesaro-Perron, and Marcinkiewicz-Zygmund integrals.
Professor Henstock develops here the general axiomatic theory of Riemann-type integration from first principles in such a way that familiar classes of integrals (such as Lebesgue and Wiener integrals) are subsumed into the general theory in a systematic fashion. In particular, the theory seeks to place Feynman integration on a secure analytical footing.
By adopting an axiomatic approach, proofs are, in general, simpler and more transparent than have previously appeared. The author also shows how one proof can prove corresponding results for a wide variety of integrals. As a result, this book will be the central reference work in this subject for many years to come.
Readership: Research workers and advanced graduate students in analysis; some mathematicians and theoretical physicists whose work touches on the Wiener and Feynman integrals
Author(s): Ralph Henstock
Series: Oxford Mathematical Monographs
Publisher: Oxford University Press
Year: 1991
Language: English
Pages: C, xii+262, B
Cover
Series Editors
Publications of OXFORD MATHEMATICAL MONOGRAPHS
The General Theory of Integration
© Ralph Henstock 1991
ISBN 0-19-853566-X
QA311. H54 1991 515' 4-dc20
LCCN 90020901
PREFACE
CONTENTS
CHAPTER 0 INTRODUCTION AND PREREQUISITES
0.1 INTRODUCTION: THE CALCULUS, LEBESGUE, AND GAUGE (KURZWEIL-HENSTOCK) INTEGRALS
0.2 BASIC DEFINITIONS FOR SIMPLE SET THEORY
0.3 BASIC DEFINITIONS FOR ALGEBRA
0.4 BASIC DEFINITIONS AND THEORY FOR TOPOLOGY
0.5 BASIC DEFINITIONS AND THEORY FOR ORDER
0.6 SHORT HISTORY OF INTEGRATION
CHAPTER 1 DIVISION SYSTEMS AND DIVISION SPACES
1.1 DEFINITIONS
1.2 THE NORM INTEGRAL (HENSTOCK (1968b, pp. 219-20, Ex. 43.1))
1.3 THE REFINEMENT INTEGRAL (HENSTOCK (1968b, p. 220, Ex. 43.2))
1.4 THE GAUGE (KURZWEIL-HENSTOCK) INTEGRAL (HENSTOCK (1968b, pp. 220, 221, Ex. 43.3; and 1988a))
1.5 THE GAUGE INTEGRAL, ASSOCIATED POINTS AT VERTICES, AND INFINITE INTERVALS
1.6 McSHANE'S MODIFICATION, GIVING ABSOLUTE INTEGRALS
1.7 SYMMETRIC INTERVALS
1.8 THE DIVERGENCE THEOREM
1.9 THE GENERAL DENJOY INTEGRAL
1.10 BURKILL'S APPROXIMATE PERRON INTEGRAL
1.11 DIVISION SYSTEMS AND SPACES IN A TOPOLOGY
CHAPTER 2 GENERALIZED RIEMANN AND VARIATIONAL INTEGRATION IN DIVISION SYSTEMS AND DIVISION SPACES
2.1 FREE DIVISION SYSTEMS, THE FREE p-VARIATION, FREE NORM VARIATION, AND CORRESPONDING VARIATIONAL INTEGRAL, AND THE FREE VARIATION SET
2.2 FREELY DECOMPOSABLE DIVISION SYSTEMS
2.3 DIVISION SYSTEMS, THE GENERALIZED RIEMANN AND VARIATIONAL INTEGRALS, THE VARIATION, AND THE VARIATION SET
2.4 DECOMPOSABLE DIVISION SYSTEMS
2.5 DIVISION SPACES
2.6 DECOMPOSABLE DIVISION SPACES AND INTEGRATION BY SUBSTITUTION
2.7 ADDITIVE DIVISION SPACES
2.8 DECOMPOSABLE ADDITIVE DIVISION SPACES
2.9 STAR-SETS AND THE INTRINSIC TOPOLOGY
2.10 SPECIAL RESULTS
CHAPTER 3 LIMITS UNDER THE INTEGRAL SIGN, FUNCTIONS DEPENDING ON A PARAMETER
3.1 INTRODUCTION AND NECESSARY AND SUFFICIENT CONDITIONS
3.2 MONOTONE SEQUENCES AND FUNCTIONS
3.3 THE BOUNDED RIEMANN SUMS TEST AND THE MAJORIZED (DOMINATED) CONVERGENCE TEST OF ARZELA AND LEBESGUE
CHAPTER 4 DIFFERENTIATION
4.1 THE DIFFERENTIATION OF STRONG VARIATIONAL INTEGRALS
4.2 FURTHER RESULTS ON DIFFERENTIATION
CHAPTER 5 CARTESIAN PRODUCTS OF A FINITE NUMBER OF DIVISION SYSTEMS (SPACES)
5.1 FUBINI-TYPE RESULTS
5.2 TONELLI-TYPE RESULTS ON THE REVERSAL OF ORDER OF DOUBLE INTEGRALS
CHAPTER 6 INTEGRATION IN INFINITE DIMENSIONAL SPACES
6.1 INTRODUCTION
6.2 DIVISION SPACE INTEGRATION
CHAPTER 7 PERRON-TYPE, WARD-TYPE, AND CONVERGENCE-FACTOR INTEGRALS
7.1 PERRON-TYPE INTEGRALS IN FULLY DECOMPOSABLE DIVISION SPACES
7.2 WARD INTEGRALS IN DECOMPOSABLE DIVISION SYSTEMS AND SPACES
7.3 CONVERGENCE-FACTOR INTEGRALS IN FULLY DECOMPOSABLE DIVISION SPACES
CHAPTER 8 FUNCTIONAL ANALYSIS AND INTEGRATION THEORY
8.1 INTRODUCTION
8.2 YOUNG'S INEQUALITY AND ORLICZ SPACES
8.3 THE FUNCTIONAL ANALYSIS OF CONTINUOUS GENERALIZED RIEMANN INTEGRALS
8.4 DENSITY INTEGRATION
REFERENCES
NAME INDEX
SUBJECT INDEX
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