Interchange differential operator with Lebesgue integral.

Question

Under what condition am I able to interchange a differential operator with an integral? More precisely, given a function $f:\Omega\times U\to\Bbb R$ from a measure space $(\Omega,\mathscr A,\mu)$ and $U\subset\Bbb R^n$ with the property $f(\cdot,y)\in L^1(\mu)$ for all $y\in U$ and $f(x,\cdot)$ is in the domain of some differential operator $D$ for all $x\in\Omega$, when is it the case that: $$ D\int_\Omega f(x,y)\,d\mu(x)=\int_\Omega Df(x,y)\,d\mu(x) $$ Thanks in advance.

Answer 1

Theorems 4 and 5 in Roussas' "An Introduction to Measure-theoretic Probability" (chapter 5 p.97) give sufficient conditions for $n=1$.