Suppose I have a function $f \in L^2(R^n)$ with compact support, and a mollifier $\phi(x) \in C^\infty(R^n)$ with compact support s.t. $\int \phi(x) dx = \int \phi_\epsilon(x) = 1$ and $lim_{\epsilon \to 0} \phi_\epsilon(x) = \delta(x)$ where $\phi_{\epsilon}(x) = \epsilon^{-n} \phi(x/\epsilon)$ and $\delta(x)$ denotes the dirac delta.
Let $f_\epsilon = f * \phi_\epsilon$, where $*$ denotes convolution.
In general, is it possible to pass the limit of the integral of $f_\epsilon$ inside and say that $\lim_{\epsilon \to 0} \int f * \phi_\epsilon = \int \lim_{\epsilon \to 0} f *\phi_\epsilon$?
And if it is not, what type of stronger condition would I need to be able to do that?
I thought about using the Dominated Convergence Theorem, but I wasn't sure how to come up with a dominating function that works for every $\epsilon$.