Integration Theory
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1078
A. J. Ell M. Janssen R van der Steen
Integration Theory
Springer-Verlag Berlin Heidelberq New York Tokyo 1984
Authors
A.J. E. M. Janssen Philips' Research Laboratories PO.Box 80.000, 5600 JA Eindhoven, The Netherlands P van der Steen Department of Mathematics, University of Technology PO. Box 513, 5600 MB Eindhoven, The Netherlands
AMS Subject Classification (1980): 28-01, 26-01 ISBN 3-540-13386-0 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-13386-0 Springer-Verlag New York Heidelberg Berlin Tokyo
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© by Springer-Verlag Berlin Heidelberg 1984 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach / Bergstr. 2146/3140-543210
PREFACE
Presenting yet another book on integration theory requires some justification. When writing the present material, we had in mind to explain what Lebesgue integration is and how it can be developed. An important point was to reconcile the various methods to introduce the integral. 11any of the ideas used occur already in papers by Stone in 1948-1950. But the form we present them in, and much else as well, springs from a series of lectures by N.G. de Bruijn, around 1964. The General Introduction extensively explains our intentions. Thanks are due to N.G. de Bruijn: without him the book would never have been written; to J.W. Nienhuys, who critically read portions of the manuscript; and to David Klarner, who read the whole manuscript and suggested many improvements. Parts of the manuscript were typed at the Mathematical Department of the Technological University Eindhoven. The final typing was done by Mrs. Elsina Baselmans-weijers, who did a superb job, as usual. A.J.E.M. Janssen P. van der Steen
CONTENTS General introduction Notation
5
Chapter O. Preliminaries
6
0.1. 0.2. 0.3. 0.4. 0.5.
6 9
Algebraic preliminaries Topological preliminaries Normed spaces and inner product spaces Summation The Riemann and the Riemann-Stieltjes integral
18
20 24
Chapter I. The fundamental stations and their connections
34
1.1. 1.2. 1.3. 1.4.
36
Basic functions, norms, and the extension process Auxiliary functions The Daniell approach The circle line
47 55
62
Chapter 2. Further development of the theory of integration
66
2.1. 2.2. 2.3. 2.4.
67 76
Measurable functions and measurable sets Further extension of the integral LP-spaces The local norm
Chapter 3. Integration on measure spaces 3.1. 3.2. 3.3. 3.4.
Semirings and measures Measure spaces and integration Measurability with respect to a a-algebra, approximation properties The measure generated by an integral
81
90
95 96 103 III
120
Chapter 4. Integration on local
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