Integration Theory (with Special Attention to Vector Measures)
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315 Klaus Bichteler The University of Texas at Austin, Austin, TX/USA
Integration Theory (with Special Attention to Vector Measures)
Springer-Verlag Berlin· Heidelberg· New York 1973
AMS Subject Classifications (1970): 28A30, 28A45, 46G15
ISBN 3-540-06158-4 Springer-Verlag Berlin' Heidelberg· New York ISBN 0-387-06158-4 Springer-Verlag New York· Heidelberg· Berlin
This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically those of translation, reprinting, re·use of illustrations, broadcasting, reproduction by photocopying machine or similar means, and storage in data banks. Under § 54 of the German Copyright Law where copies are made for other than private use, a fee is payable to the publisher, the amount of the fee to be determined by agreement with the publisher.
© by Springer-Verlag Berlin' Heidelberg 1973. Library of Congress Catalog Card Number 72-97636. Offsetdruck: Julius Beltz, Hemsbach IBergstr.
PREFACE These notes reflect a course on vector measures given at The University of Texas at Austin in the summer term of 1970.
They contain
as much material as is usual for a course on this topic. The presentation is unusual, though, and might possibly be of some interest even to the expert.
It is motivated by the desire to carry
M. H. Stone's unified treatment of set functions and Radon measures as far as possible. This has been achieved by the use of a new and, I believe, natural definition of measurability, far-reaching enough to embrace the definitions both of Bourbaki and Caratheodory-Halmos and combining the flexibility and intuitive appeal of the former with the applicability to probability theory of the latter.
It can also be applied to more general situations
than either of these. Another unorthodox feature is
the axiomatic treatment of upper
integrals and their generalizations, the upper gauges, leading to the Daniell-integral also for measures and linear maps that do not have finite variation. I should like to express my gratitude to Professor Roy P. Kerr, who eradicated a frightful number of mistakes and Germanisms from the original manuscript, and to NSF for their support of these notes; and last but not least to Ms. Linda S. White who typed these notes with extreme skill and diligence and with inexhaustible patience. Klaus Bichteler
*) Supported
by NSF Grant GP-?0541
*
TABLE OF CONTENTS
1.
Introduction. • . .
CHAPTER I. 2. 3. 4. 5. 6.
7. 8. 9.
12. 13. 14. 15. 16. 17.
18. 19. 20.
21. 22. 23. 24.
94
104 116
129 138
143 147
156 161
MEASURABILITY MEASURABLE FUNCTIONS AND SETS
(ffi,M)-measurability • • • . . • . Integrability and measurability criteria The spaces and Li . • • . . . .
£i
§2.
73 83
VARIOUS NEW UPPER GAUGES
The p-norms (1 < P « "") Essential upper-gauges Smooth upper gauges The upper integral AM Strictly localizable upper gauges Bauer's theory . . . • •
CHAPTER III. §1.
51 65
INTEGRATION OF VECTOR-VALUED FUNCTIONS
Extension under an upper norm Extension of linear maps §3.
12 24
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