Topology and Measure
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133 Flemming Tops0e Department of Mathematics, University of Copenhagen
Topology and Measure
Springer-Verlag Berlin· Heidelberg· NewYork 1970
Lecture Notes in Mathematics A collection of informal reports and seminars Edited by A. Dold, Heidelberg and B. Eckmann, ZOrich
133 Flemming Tops0e Department of Mathematics, University of Copenhagen
Topology and Measure
Springer-Verlag Berlin· Heidelberg· NewYork 1970
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 1970. Library of Congress Catalog Card Number 75-120379. Printed in Germany. Tide No. 3289.
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CONTENTS Preface introduction acknowledgments Preliminaries
IV IX
PART I 1. Measure and integral, definitions 2. Basic result on construction of a measure 3. Basic result on construction of an integral 4. Finitely additive theory 5. From "Baire" measures to "Borel" measures, an abstract approach 6. Construction of measures by approximation from outside and by approximation from inside 7. On the possibility of prOViding a space of measures with a vague topology
1 3 6 15 21
26 31
PART II 8. Definition and basic properties of the weak topology 9. Compactness in the weak topology 10. Criteria for weak convergence 11. On the structure of
40 42 45
12. 13. 14. 15. 16. 17.
51 54 60
A problem related to questions of uniformity First solution of the e-problem Second solution of the e-proolem Uniformity classes Joint continuity Preservation of weak convergence
47
64 66 68
Notes and remarks
72
References
78
PREFACE INTRODUCTION
Below we shall comment on the development which led to the results of the present volume (and of [26]). It will be seen that our investigations took their starting point in the theory of weak convergence of (probability-) measures, and that the results on measure and integration theory to be found in part I emerged as a kind of "by-product". During the inspiring lectures of professor P. Billingsley, I was for the first time presented to the theory of weak convergence. These lectures, which were based on the manuscript to the book
[3],
took
place in 1964-65 while professor Billingsley visited the institute of math. statistics at the university of Copenhagen. The contact with Billingsley resulted, among other things, in the joint paper
[4].
This paper was the
starting point of the development leading to the results obtained in sections 12-17. The academic year 1965-66 I spent at the statistical laboratory, the university of Cambridge, England. There I met professor K.R. Parthasarathy, who was at that time working on his book
dealing with weak
convergence too. Once
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