Measurability of level sets of measures Let $X$ be a standard Borel space, and $\mathcal P(X)$ be the set of Borel probability measures on $X$ with a topology of weak convergence. It is known that $\{p:p(B) = 1\}$ is a Borel subset of $\mathcal P(X)$ for any Borel set $B\subset X$ we have. Furthermore, such set is analytic whenever $B$ is.
I wonder, whether $\{p:p(B) = 1\}$ is analytically/universally measurable given that $B$ is analytically/universally measurable.

All these facts can be found in the book by Bertsekas and Shreve "Stochastic Optimal Control". In particular, the remark before Proposition B.12 in the appendix contains all references to the fact that $\{p:p(B) = 1\}$ is analytic or Borel/analytically/limit/universally measurable given that $B$ is analytic or Borel/analytically/limit/universally measurable respectively. Respecting the order: Proposition 7.43, Proposition 7.25, Proposition 7.43, Proposition B.12 and Corollary 7.46.1.
