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


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.


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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