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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 $\mathbb P_n(A)\not \rightarrow \mathbb P(A)$.

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The first part in contained in the statement of portmanteau theorem. One direction is not specially hard (approximate pointwise the characteristic function of a closed set by a continuous function in order to get $\limsup_n\mathbb P_n(F)\leqslant \mathbb P(F)$).

For the second one, the question reduces to the following: if $f_n,f$ are non-negative integrable function of integral $1$ and for each continuous $\phi$, we have $\int_{\mathbb R}f_n(x)\phi(x)\mathrm dx\to\int_{\mathbb R}f(x)\phi(x)\mathrm dx$ can we extend this to $\phi\in L^\infty$? (this follows from an approximation argument). The answer is no.

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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.

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