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Choquet theorem for compact measures. (English) Zbl 0577.28009

The purpose of the paper is to topologize the set of compact probability measures \({\mathcal P}\) on a countably generated measurable space (X,B) in such a way that its closed convex subsets posses the Choquet type integral representation property. The theory presented here states the property for sets such as \[ H_{T,C}=\{p\in {\mathcal P}| t(p)=p,\quad t\in T,\quad (\int_{X}g dp,\quad g\in D)\in C\}, \] where T is a countable set of measurable maps from X into itself, D is a countable set of real measurable functions and \(C\subset R^ D\) is a closed convex set. The paper extends the earlier results by H. v. Weizsäcker and G. Winkler [Math. Ann. 246, 23-32 (1979; Zbl 0403.46015)] and by A. Maitra [Trans. Am. Math. Soc. 229, 204-225 (1977; Zbl 0357.28020)].

MSC:

28C05 Integration theory via linear functionals (Radon measures, Daniell integrals, etc.), representing set functions and measures
46A55 Convex sets in topological linear spaces; Choquet theory
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