Another interesting property concerning the probability measures on the rationals

Abstract:

Let $X$ be a perfect, complete, separable metric space and $P(X)$ denote the space of Borel probability measures on $X$ equipped with the topology of weak convergence. If $Y$ is a countable dense subset of $X$ then $P(Y)$ is not a ${G_{\delta \sigma }}$ subset of $P(X)$. Furthermore if $X$ is separable, complete and metric, and $Y \subseteq X$, and $P(Y)$ is a ${G_{\delta \sigma }}$ subset of $P(X)$, then $P(Y)$ is in fact a ${G_\delta }$ subset of $P(X)$.