Choquet theory in metric spaces (Q1578899)

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 \[ \varphi(z)\leq \int_{\text{ext}(K)}\varphi(t) d\mu_z(t),\quad \varphi\in C_c(M). \]