I apologise in advance if my question is too basic.

Some notation:

$(X,\cal{X})$ denotes a measurable metric space where $X$ is a metric space and $\cal{X}$ is the associated Borel sigma algebra.

$B(X)$ is the space of all bounded continuous functions defined on $X$.

Let $\{\mu_n\}$ and $\{\nu_n\}$ be sequences of probability measures on the above measurable space $(X, \mathcal{X})$. Assume that each $\mu_n$ is absolutely continuous with respect to $\nu_n$, with an density $h_n\in B(X)$. Suppose that $\mu_n\to \mu$, $\nu_n\to \nu$ in the weak star topology and $h_n$ converges to a bounded continuous function $h$.

**Question:**
I would like to know if $\mu\ll\nu$. If so, is $h$ the density? If not, is there some condition in order
to have $\mu\ll\nu$?

Other information that can be useful is that each $\nu_n$ and $\mu_n$ has support in a compact subset $K_n\subset X$ which increase to $X$, i.e, $X=\bigcup K_n$.

**Edit:** $h_n$ converges to $h$ uniformly in compacts sets