Let $X$ and $Y$ be Polish spaces, let $\mathcal{P}(Y)$ be the space of Borel probability measures on $Y$ endowed with the smallest $\sigma$-algebra such that all functions of the form $\nu\mapsto\nu(A)$ with $A\subseteq Y$ measurable are measurable, and let $g:X\to\mathcal{P}(Y)$ be a measurable function. If there exists a probability measure $\mu$ on $Y$ such that $g(x)$ is absolutely continuous with respect to $\mu$ for all $x\in X$, then there exists a jointly measurable function $h:X\times Y\to\mathbb{R}$ such that $h(x,\cdot)$ is a Radon-Nikodym derivative of $g(x)$ with respect to $\mu$ for all $x\in X$. This follows, for example, from [Paul André Meyeyer 1966, Probability and Potentials, Result VIII-10].
I would like to know if there is some reasonable condition on $g$ that is equivalent to, or at least implies, the existence of such a function $h:X\times Y\to\mathbb{R}$ continuous in $X$. If needed, one may assume $X$ to be compact.
Such continuously parametrized density functions are used in various areas of mathematical game theory, and I would like to know if there is a less ad hoc way to think about them.
