Probability measure functors preserving the ANR-property of metric spaces

Abstract:

Let ${P_k}\left ( X \right )$ denote the set of all probability measures on a metric space $X$ whose supports consist of no more than $k$ points, equipped with the Fedorchuk topology. We prove that if $X \in {\text {ANR}}$ then ${P_k}\left ( X \right ) \in {\text {ANR}}$ for every $k \in {\mathbf {N}}$. This implies that for each $k \in {\mathbf {N}}$ the functor ${P_k}$ preserves the topology of separable Hilbert space.