monotone convergence of $f_n$ plus weak convergence of $\mu_n$ implies convergence?

Question

I would like to know if somebody is aware of some result that looks like the following.

Let us consider the space $C_b(X)$ of continuous bounded function over a measurable space $X$.

Suppose that:

1. $f_n \uparrow_n f$, i.e. $(f_n)$ converge monotonically towards $f$, and moreover $f_n\geq 0 \,\forall n$.
2. Suppose $\mu_n\xrightarrow{w*}\mu$, with $(\mu_n)$ and $\mu$ $\sigma$-additive measures over $X$ (where the convergence is w.r.t. the weak-* topology i.e. $\mu_n\xrightarrow{w*}\mu$ iff $\forall g\in C_b(X), \int g d\mu_n \rightarrow \int g d\mu$).

Do we have that $\int f_n d\mu_n \rightarrow \int f d\mu$?