In finance there is the question, of what conditions would justify the following step:
$$\partial_S E(\max(S-K,0))=E(\partial_S\max(S-K,0))$$
where $\partial_S$ stands as the partial derivative in the distribution sense with respect to S, where in this case $\max(S-K,0)$ would be the distribution functional.
If I generalize the question into:
$$\partial_S E(g(S))=E(\partial_S g(S))$$
What would I have to require from $g$, for the shown step to hold?
I know that in the case of the usual derivative, the partial derivative can go inside the integral if the function is Lipschitz continuous, using the denominated convergence theorem.
But regarding distributions, I could not find any proposition that would back this move.
I really appreciate any help you can provide!