Let $(\mathcal{X},\Sigma,P)$ be a Polish probability measure space, and $(\mathcal{X}^n,\Sigma^{\otimes n},P^n)$ be the product of its $n$ copies. Let $t: x^n \in \mathcal{X}^n \mapsto L_{x^n} \in \mathcal{P}(\mathcal{X})$ be the empirical measure function, where $L_{x^n}$ is the empirical measure of the $n$-length sequence $x^n$, and $\mathcal{P}(\mathcal{X})$ is the set of probability measures on $(\mathcal{X},\Sigma)$. We consider two kinds of subsets of $\mathcal{P}(\mathcal{X})$:

- A subset $A$ of $\mathcal{P}(\mathcal{X})$ such that $t^{-1} (A)$ is $\Sigma^{\otimes n}$-measurable.
- A subset $B$ of $\mathcal{P}(\mathcal{X})$ such that $B$ is measurable with respect to the Borel $\sigma$-algebra induced by the weak topology.

Obviously, $B$ is a special case of $A$. My question is: For any $A$ above, can we find a sequence $B_k$ as the above such that $P\circ t^{-1} (B_k)\to P\circ t^{-1} (A)$ as $k\to \infty$? Are there references for this question? Thanks.