Reference request: Interchange Inf and integral

Question

I'm looking for a reference to a result apparently by Rockafellar which says something like if $f(x,y)$ is measuable in $x$ and continous in $y$ and $x \in X$ a "sufficiently nice" subspace of a finite-measure space $(\Omega,\mathfrak{F},\mu)$ then $$ \inf_{x \in X} \int f(s,x(s))\mu(ds) = \int \inf_{x \in \mathbb{R}^d}f(s,x)\mu(ds)? $$ I might be missing conditions (obviously wrt "sufficiently nice").

Please help with reference, I have a probability background so a "not too fancy" but short read would be ideal.

Answer 1

This article contains everything you've ever recently dreamed of and more :) In particular see the first theorem for precisely what you're looking for + more modern formulation!