Let us consider a bivariate random variable $X\in \mathbb{R}^2$ with $pdf$ $f$. Also let, based on a sample of size $n$, let the the estimator of the density be $f_n(x)$ at $x\in \mathbb{R^2}$ and we know that $f_n(x)\rightarrow f(x)$, a.s uniformly.
Now, we consider a continuous map $\phi:\mathbb{R^2}\rightarrow[0,1]$.
Then how to show $\int_{\mathbb{R^2}}\phi(x)f_n(x)dx\rightarrow\int_{\mathbb{R^2}}\phi(x)f(x)dx$, a.s.?
I am trying to prove this using DCT, but the problems are: How to extend the DCT to the multivariate case and what could be the dominated function (since $f$ could be unbounded function)?
Thanks in advance.