A criterion for weak convergence of probability measures 
Let $\mathbb P_n$ and $\mathbb P$ be probability measures. We have that $\mathbb P_n$ converges weakly to $\mathbb P$ if for each continuous bounded function, $\int f(x)\mathrm d\mathbb P_n\to\int f(x)\mathrm d\mathbb P$. Show that $\mathbb P_n$ weakly converges to $\mathbb P$ if and only if $\lim_{n\rightarrow \infty} \mathbb P_n(A) = \mathbb P(A)$ for all Borel set $A$ with $P(\partial A)=0.$

And here is a related question, which I couldn't solve as well. 

Give an example of a family of probability measures $\mathbb P_n$ on ($\mathbb{R}, \mathcal{B}(\mathbb{R}$)) which weakly converges to $\mathbb P$, all of $\mathbb P_n$ and $\mathbb P$ absolutely continuous with respect to the Lebesgue measure, yet there is a Borel set $A$ such that $\mathbb P_n(A)\not \rightarrow \mathbb P(A)$.


This is the standard equivalence between weak convergence and convergence in genereal. Please refer to Chapter III of Probability by A. Shiryaev. 
For the second question, I can only come up with examples similar to what Davide mentioned.
