An example in Billingsley's book on weak convergence of measures is like this:
Let $P_n$ be point mass at $x_n$ and $P$ be point mass at $x$. $P_n \Rightarrow P$ is seen to be equivalent to $x_n \to x$.
Let $A = \{x_2,x_4,x_6,\ldots\}$. Weak convergence implies $\lim P_n (B) = P(B) $ for all continuity sets of $P$. Clearly the set $A$ does not contain $x$ and hence is a continuity set for $P$. Then why does $P_n(A)$ doesnot converge to any limit?
