# Weak convergence of absolutely continuous probability measures

Suppose I have a sequence of probability measures $P_n$ which converges weakly to a probability measure $P$. Then we know that $\lim_{n}$ $P_n$ $(A)$ $=$ $P(A)$ if $A$ is a $P$-continuity set.
Is it true that if, in addition we know that each $P_n$ and $P$ are absolutely continuous, then $\lim_{n}$ $P_n$ $(A)$ $=$ $P(A)$ for any $A$?