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Some properties of maximal measures on compact convex sets

Published online by Cambridge University Press:  24 October 2008

C. J. K. Batty
C. J. K. Batty
Affiliation:
Department of Mathematics, University of Edinburgh, Edinburgh EH9 3JZ
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Abstract

Let be a maximal measure on a compact convex set K, K* be the state space of the space of all continuous functions f: KK ℝ which are affine in the first variable, 1 be the -algebra on K generated by the Baire sets and the compact extremal subsets of K, and = {BeK1}. Then

(i) For any fixed continuous function g:K ℝ and -almost all x in K, there is a closed face of K containing x on which g is constant.

(ii) The image of under the map :KK* defined by f, (x) = f(x, x) is the unique maximal measure on K* representing its barycentre

(iii) induces a measure on (eK) satisfying certain regularity conditions.

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 94 , Issue 2 , September 1983 , pp. 297 - 305
Copyright
Copyright © Cambridge Philosophical Society 1983

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