Choquet Order and Simplices with Applications in Probabilistic Model
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1145
Gerhard Winkler
Choquet Order and Simplices with Applications in Probabilistic Models
Springer-Verlag Berlin Heidelberg New York Tokyo
Author
Gerhard Winkler Mathematisches Institut, Universitiit Munchen Theresienstr. 39, 8000 Munchen 2, Federal Republic of Germany
Mathematics Subject Classification (1980): primary: 46A55 secondary: 18B99, 28C99, 46E27, 52A07, 60G05 ISBN 3-540-15683-6 Springer-Verlag Berlin Heidelberg New York Tokyo ISBN 0-387-15683-6 Springer-Verlag New York Heidelberg Berlin Tokyo
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 "Verwertungsgesellschaft Wort", Munich.
© by Springer-Verlag Berlin Heidelberg 1985 Printed in Germany Printing and binding: Beltz Offsetdruck, Hemsbach/Bergstr. 2146/3140-543210
PREFACE
These lecture notes present a summary of certain properties of Choquet order on locally convex spaces, an examination of inverse systems of simplices and applications in probability theory. The central results were obtained by the author in 1982 and 1983. There is little or no overlap with recent surveys - like R.D. Bourgin's monograph (1983). The reader I had in mind when writing down the
is a student of
mathematics attending a seminar in the last third of his study. So the proofs are rather thorough and the specialist will probably skip a lot of details. The reader is assumed to be familiar with the basic ideas of topological measure theory and locally convex vector spaces. It is useful but not necessary to have some familiarity with the Choquet theory. I am indebted to J.P.R. Christensen, S. Dierolf, G. Godefroy, Chr. Hele, H. Kellerer, E. Kolb, R. Kotecky, Z. Lipecky, D. Preiss and last but not least H.v. Weizsacker for their help and useful comments.
CONTENTS
INTRODUCTION CHAPTER O.
NOTATIONS, DEFINITIONS AND CONVENTIONS
CHAPTER 1.
BASIC CONCEPTS FROM NONCOMPACT CHOQUET THEORY
1.1.
6
13
Barycenters, representing measures and the barycenter map
13
1. 2.
Measure convex sets
21
1. 3.
Choquet order
28
1.4.
Boundary measures
.
CHAPTER 2. 2.1.
Simplices
.••••••..••..•••••.•••....••••.
38
•....••.....•.•............•••.....•...
46
54
FOUR ASPECTS OF CHOQUET ORDER Measures smaller in Choquet order live on smaller sets
2.2.
•.••..•..••....•.......•........••.
..••.•.••.••••.••••......••..•.•••••
Uniform tightness of sets of measures bounded from above in Choquet order; convex tight measures
2.3.
CHAPTER 3.
57
Extension of tight measures from the weak Borel 0algebra to the strong Borel o-algebra
2.4.
54
.••••.••••.
62
Nets of measures monotone in Choquet order
68
INVERSE LIMITS OF SIMPLICES
75 ••.•••••••.•••••••.
76
..••..•.•.•....••••••
79
3.1-
The intersection of simplices
3.2.
Inverse
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