So I have a sequence of estimators $\phi_n$ that converges almost surely to $\phi^*$. These estimators are used to parametrise a function $f_{\phi_n}(x)$ which thus forms a random process to be integrated:
$I_n= \int_X f_{\phi_n}(x) dp(x)$.
Now, I can show that there exists a real $M$ such that $\vert f_{\phi}(x)\vert \le M$ for all $x\in X$ and $\phi\in\Phi$. As $f$ is continuous in $\phi$, I have $f_{\phi_n}(x)\xrightarrow{N\rightarrow\infty}f_{\phi^*}(x)$ almost surely. Can I apply an analogue of the dominated convergence theorem, i.e something like
$I_n\xrightarrow{N\rightarrow\infty} \int_X f_{\phi^*}(x) dp(x)$ almost surely?