Suppose that $\{\mu\_k\}$ is a sequence of probability measures defined on $(I,\mathcal{B})$, where $I$ is a closed real interval and $\mathcal{B}$ is its Borel $\sigma$-algebra, such that:


* each $\mu\_k$ is made of countably many atoms $\{p^n\_k\}$;
* for every $k$, the set $\bigcup\_{n\in\mathbb{N}}p^n\_k$ is not dense in $I$;
* $\bigcup\_{n,k\in\mathbb{N}}p^n\_k$ is dense in $I$.


I wonder which further assumptions are needed to ensure that:


* $\mu\_k$ weakly converges to a probability measure $\mu$,
* $\mu$ is absolutely continuous.


Thanks a lot!



Convergence of atomic measures with countably many atoms
(I cannot make any comment so I have no choice but post my comment as answer)  

I'm not sure which kind of $\mu\_k$ you want to control. You are surely given tightness for free, so mainly if you can describe the limit law by some total set of linear functionals then your work is done.  

I think you know there are a lot of possible choices like moment generating function, characteristic function, cumulative distribution function.  

Or you may try to prove the convergence of moments, or of Stieljes transform.  

Or, you may craft one yourself based on properties of your $\mu\_k$.  

Hope this help.


