Weak convergence of absolutely continuous probability measures

Hi there,

Suppose I have a sequence of probability measures $\displaystyle P_n$ which converges weakly to a probability measure $\displaystyle P$. Then we know that $\displaystyle \lim_{n}$$\displaystyle P_n$$\displaystyle (A)$$\displaystyle = $$\displaystyle P(A)$ if $\displaystyle A$ is a $\displaystyle P$-continuity set.

Is it true that if, in addition we know that each $\displaystyle P_n$ and $\displaystyle P$ are absolutely continuous, then $\displaystyle \lim_{n}$$\displaystyle P_n$$\displaystyle (A)$$\displaystyle =$$\displaystyle P(A)$ for any $\displaystyle A$?

Thank you very much,

Ilan