Let $X,Y$ be a standard Borel spaces (a Borel subset of a complete separable metric space), and let $\mathcal B(X),\mathcal P(X)$ denote collection of Borel sets and Borel probability measures on $X$ respectively, the latter being endowed with a topology of weak convergence. Consider a set $J\subseteq X\times \mathcal P(Y)$ such that section $J_x\neq\emptyset$ for any $x\in X$ and suppose that $J$ is an analytic set.
For a product measure $P\in \mathcal P(X\times Y)$ by $P_X$ we denote its marginal on $X$, and by $\frac{\mathrm dP}{\mathrm dP_X}$ the corresponding conditional distribution (stochastic kernel on $Y$ given $X$). Let's say that $P$ is $J$-feasible whenever $$ P_X\left(x\in X:\frac{\mathrm dP}{\mathrm dP_X}(x)\in J_x\right) = 1. $$ Can I show that the set of all $J$-feasible measures is analytic? I believe, the solution can be found if the answer to that question of mine is affirmative.