Suppose that $(S,\Sigma)$ is a measurable space with $S$ Polish and $\Sigma$ its Borel sigma algebra. Let $\mathcal{C}$ be the collection of discrete probability measures on $S$ having countably infinite many distinct point masses. For any $P \in \mathcal{C}$ there exists a set $F_P \in \Sigma^{\infty}$ of full $P^{\infty}$ measure with the property that for any $(s_1,s_2,...) \in F_P$ we have that $s_i$ is a point mass of $P$ for all $i$, and that the empirical probability measures for $(s_1,s_2,...)$ converge setwise (sometimes called *strongly*) to $P$. Let

$$G = \bigcup_{P \in \mathcal{C}} F_P $$

My question is this: is $G \in \Sigma^{\infty}$?

Additionally, is $G \in \Sigma^{\infty}$ if we change $\mathcal{C}$ to be the collection of *all* discrete probability measures on $S$?

Many thanks for any help on this.

allI mean those discrete $P$ having finiteorcountably many distinct point masses. $\endgroup$ – shanex Jun 22 '17 at 22:32