In general, my problem can be formulated as follows: Let $X$ be a random variable with value in $\mathbb R^2$, and let $G:\mathbb R^2 \times \mathbb R\rightarrow \mathbb R$ be a function which is continuous in the first argument and measurable in the second(i.e., a Caratheodory function). Assume the partial maximization $x\mapsto \sup_yG(x,y)$ is measurable. I am considering the values given by
\begin{equation} V_1=\sup\{\mathbb E(G(X,y(X))| y:\mathbb R^2\rightarrow \mathbb R \: measurable\} \\ V_2=\mathbb E(\sup_{y\in\mathbb R}G(X,y)) \end{equation} I would like to know if these two values are the same or not, assuming problem $V_1$ admits a maximizer.
In my original problem, the function $G$ is given by $G(x,y)=(x_1-f(y))(y-x_2)$, where $f:\mathbb R \rightarrow \mathbb R$ is a Borel measurable function. It seems that continuity of $G$ with respect to $y$ is quite crucial for the argmax correspondence $\Phi(x)=\{y:G(x,y)=sup_yG(x,y)\}$ to have a measurable selector. Unfortunately, this is something I do not have. I would pretty much like to have a positive answer(i.e.,$V_1=V_2$), but a counter-example will be equally appreciated! Thanks!