Let $(\Omega, \mathcal{G}, \mathbb{P})$ be a (complete) probability space, let $D$ be a compact, separable, metrizable topological space and let $$ S : \Omega \times D \rightarrow \mathbb{R}$$ be a $(\mathcal{G}\otimes\mathcal{B}(D))/\mathcal{B}(\mathbb{R})$-measurable function.
Under which conditions exists a function $$ \hat{S} : \Omega \times D \rightarrow \mathbb{R}$$ with the following two properties:
There is a set $A \in \mathcal{G}$ of measure $\mathbb{P}(A) = 1$, such that, for all $\omega \in A$, it holds that, for all $\pi \in D$, we have $S(\omega, \pi) = \hat{S}(\omega, \pi)$.
For all $\omega \in \Omega$, the map $D \ni \pi \mapsto \hat{S}(\omega, \pi) \in \mathbb{R}$ is continuous.