Let be $S$ a metric space and $X=C_b(S)$ the set of all bounded continuous functions, then it's topological dual $X^{\star}=rba(S)$ is the set of all regular Borel additive measures endowed with the variation norm. Denote by $\mathscr{P}(X)$ the subset of $rba(X)$ of all additive probability measures.
For a $G_{\delta}$ space I meant a space in which every closed set is a $G_{\delta}$ set.
Question: Is $\mathscr{P}(X)$ endowed with the weak topology a $G_{\delta}$ space? If not, is at least the singletons of $\mathscr{P}(X),$ $G_{\delta}$ sets?