Before revision:  We prove that there exists a  set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | U ) is pointwise continuous from below and continuous from above on the continuous functions.
After revision:  We prove that there exists a set-valued mapping \mathcal{P} _V from U} from \Omega } to the set of probabilities on V with compact convex values and analytic graph such that \mathcal{E}(X| U)(u)(\omega} )= \sup_{P\in\mathcal{P} _V ( u )} \int_V X(u,v)\,P(dv) if and only if \mathcal{E}(\cdot | \mathcal{G ) is pointwise continuous from below and continuous from above on the continuous functions.