Choquet theory in metric spaces
type: Item
instance of: scholarly article
title: Choquet theory in metric spaces
zbMATH Open document ID: 0966.46006
DOI: 10.4171/ZAA/952
author: T. Okon
published in: Zeitschrift für Analysis und ihre Anwendungen
publication date: 2001-08-20T00:00:00Z
full work available at URL: https://eudml.org/doc/48841
review text: Let \((M,d)\) be a matric space endowed with metric structure, i.e., a function \(E: [0,1]\times M\times M\to 2^M\) satisfying for each \(a,b\in M\), \(\lambda\in[0,1]\) and \(x\in E(\lambda, a,b)\): \[ d(x,u)\leq (1-\lambda) d(a,u)+ \lambda d(b,u)\quad (u\in M). \] Using \(E\) to introduce the notions of convex subsets in \(M\) and extreme points of these subsets, the following generalization of the classical Choquet theorem is given: Theorem. Let \(\emptyset\neq K\subseteq M\) be convex compact subset of \(M\). For all \(z\in K\) there is a probabilistic measure \(\mu_z\) on \(K\), such that \[ \varphi(z)\leq \int_{\text{ext}(K)}\varphi(t) d\mu_z(t),\quad \varphi\in C_c(M). \]
Mathematics Subject Classification ID: 46A55
Mathematics Subject Classification ID: 52A07
Mathematics Subject Classification ID: 52A30
Mathematics Subject Classification ID: 52A40
Mathematics Subject Classification ID: 52A41
Mathematics Subject Classification ID: 52A55
zbMATH DE Number: 1501858
zbMATH Keywords: metric structure
zbMATH Keywords: convex subsets
zbMATH Keywords: extreme points
zbMATH Keywords: Choquet theorem
reviewed by: Yuri A. Brudnyi
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1501858