Integrating a generalized function w.r.t. a parameter

Question

I would like to know how (and under which conditions) one can show that integration commutes with a generalized function, i.e. if $f \in \mathscr{D}'$ (or even better, if $f \in \mathscr{E}'$), under which conditions does the equality $$ \int_{\Omega} f\left( \phi(\cdot,y) \right) dy = f \left( \int_{\Omega} \phi(\cdot,y) dy \right) $$ hold?

Answer 1

Given $\phi\in C_{0}^{\infty}({\bf{R}}^{n}\times{\bf{R}}^{m})$, if we look at the Riemann sums of $y\rightarrow\phi(\cdot,y)$ on $\Omega$, they converge in $\mathcal{D}$ to $\displaystyle\int_{\Omega}\phi(\cdot,y)dy$, by linearity of $f$, the equality holds.