Topology of spaces of probability measures

Abstract

We study the space of Radon probability measures on a metric space and its subspaces , and of continuous measures, discrete measures, and finitely supported measures, respectively. It is proved that for any completely metrizable space , the space is homeomorphic to a Hilbert space. A topological classification is obtained for the pairs , and , where is a metric compactum, an everywhere dense Borel subset of , an everywhere dense -set of , and an everywhere uncountable everywhere dense Borel subset of of sufficiently high Borel class. Conditions on the pair are found that are necessary and sufficient for the pair to be homeomorphic to .