Before revision:  if and only if \mathcal{E}(\cdot |U) } is pointwise continuous from below and continuous from above on the continuous functions if and only if a dual representation of \mathcal{E_t in terms of conditional expectations minus the convex conjugate of \mathcal{E}_t holds true, where the conjugate is lower semianalytic with pointwise weakly compact level sets.
After revision:  if and only if \mathcal{E}(\cdot |U) } is pointwise continuous from below and continuous from above on the continuous functions _t in terms of conditional expectations minus the convex conjugate of \mathcal{E}_t holds true, where the conjugate is lower semianalytic with pointwise weakly compact level sets.