I'm trying to show that given two regular Borel measures on a compact space $X \subset \mathbb{R}$, $\mu_1, \mu_2$, and a bounded Borel function $f: X \to \mathbb{C}$, there exists a sequence $\{f_n\} \subset C(X)$ of continuous functions s.t $\int_X f_n d\mu_1 \underset{n \to \infty}{\to} \int_X f d\mu_1$ and $\int_X f_n d\mu_2 \underset{n \to \infty}{\to} \int_X f d\mu_2$.
My attempt:
Define $\mu = \mu_1 + \mu_2$, so that $\mu$ is regular Borel measure.
We know that there exists $\{f_n\} \subset C(X)$ s.t $\mu(\{x \in X : lim_n f_n(x) \neq f(x)\}) = 0$.
It follows that $f_n \underset{point-wise}{\to} f$ a.e$[\mu_i]$ for $i = 1,2$.
If I can choose $f_n$ so that $\{||f_n||_\infty\}$ is bounded, my question will follow from the dominated convergence theorem.
Can such a sequence be chosen, despite that we only have a.e point-wise convegence to the bounded function $f$? If so, is the rest of the argument ok? If not is there another method to show this?