I apologise in advance if my question is to basic.

Some notations 

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

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


Let $\{\mu_n\}$ and $\{\nu_n\}$ sequence of 
probability measures in the above  
mensurable  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 $\mu<<\nu$, in case of 
positive answer, is $h$ the density. In case of 
negative answer, there is some condition in order 
to have $\mu<<\nu$?

Other information that can be usefull
 is that each $\nu_n$  and $m_n$ has support in a 
compact subset $K_n\subset X$ which one 
increases to $X$, i.e, $X=\bigcup K_n$.