Topology of spaces of probability measures

T. O. Banakh ; T. N. Radul

Sbornik. Mathematics, Tome 188 (1997) no. 7, pp. 973-995

Résumé

We study the space $\widehat P(X)$ of Radon probability measures on a metric space $X$ and its subspaces $P_c(X)$, $P_d(X)$ and $P_\omega (X)$ of continuous measures, discrete measures, and finitely supported measures, respectively. It is proved that for any completely metrizable space $X$, the space $\widehat P(X)$ is homeomorphic to a Hilbert space. A topological classification is obtained for the pairs $(\widehat P(K),\widehat P(X))$, $(\widehat P(K),P_d(Y))$ and $(\widehat P(K),P_c(Z))$, where $K$ is a metric compactum, $X$ an everywhere dense Borel subset of $K$, $Y$ an everywhere dense $F_{\sigma \delta }$-set of $K$, and $Z$ an everywhere uncountable everywhere dense Borel subset of $K$ of sufficiently high Borel class. Conditions on the pair $(X,Y)$ are found that are necessary and sufficient for the pair $(\widehat P(X),P_\omega (Y))$ to be homeomorphic to $(l^2(A),l^2_f(A))$.