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.



Measurability of a particular set generated by discrete probability measures
Define $\bar f\_n(x)=\limsup\_{N\to\infty} \frac 1N\#\{j\le N\colon x\_j=x\_n\}$,
$\underline f\_n(x)=\liminf\_{N\to\infty} \frac 1N\#\{j\le N\colon x\_j=x\_n\}$, $g\_n(x)=\prod\_{j<n}\mathbf 1\_{\{x\_j\ne x\_n\}}$.


Then define $F\_n(x)=g\_n(x)\cdot \mathbf 1\_{\bar f\_n(x)=\underline f\_n(x)}\cdot \bar f\_n(x)$.
This function is the limiting density of terms that are equal to $x\_n$ provided that (1) $x\_n$ is the first time this point has appeared; (2) the limiting density of $x\_n$'s exists.


Then 
$$
G=\left\{x\colon \sum\_{n=1}^\infty F\_n(x)=1\right\}\cap \bigcap\_n\{x\colon \bar f\_n(x)>0\}.
$$ 


