I apologise in advance if my question is too basic.

Some notation:

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

2. $B(X)$ 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