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

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 $\int f_n d\mu_n \rightarrow \int f d\mu$?

• Presumably these are finite signed measures? Or are they finite positive measures? – Robert Israel Nov 4 '14 at 18:11
• Sorry for not specifying.. The measures are finite. – Lorenzo Bastianello Nov 4 '14 at 22:33
• You mean continuous function on a topological space? If the space is compact then we can use Dini's theorem. If the space is metrizable, we can use a continuous function which takes the value $1$ on a well chosen compact set. – Davide Giraudo Nov 6 '14 at 10:43
• In fact I am interested in $X=\mathbb{N}$ and therefore $C_b(X)=l^{\infty}$. – Lorenzo Bastianello Nov 6 '14 at 15:08