  For expectation functions on metric spaces, we provide sufficient conditions
for epi-convergence under varying probability measures and integrands, and
examine applications in the area of sieve estimators, mollifier smoothing,
PDE-constrained optimization, and stochastic optimization with expectation
constraints. As a stepping stone to epi-convergence of independent interest, we
develop parametric Fatou's lemmas under mild integrability assumptions. In the
setting of Suslin metric spaces, the assumptions are expressed in terms of
Pasch-Hausdorff envelopes. For general metric spaces, the assumptions shift to
semicontinuity of integrands also on the sample space, which then is assumed to
be a metric space.
