Let $X$ be a Polish space and $(P_n)$ a sequence of Borel probabilities which converges weakly in measure to a Borel probability $P$. By this i mean that for any $f\in C_b(X)$ which is continuous and bounded $$\lim \int_X f(x) P_n(dx)= \int_X f(x) P(dx)$$ I further assume that there is a $A$ such that for any $n$ A is $P_n-$negligible i.e. for any $n\in \mathbb{N}$ there is a Borel set $B_n$ with $$P_n(B_n) =0$$ and $$A\subset B_n$$ My question is the following : does it imply that $A$ is $P-$negligible i.e. that i can find a Borel set $B$ with $A\subset B$ and $$P(B)=0$$ ???

## 1 Answer

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The answer is obviously "no" (take for $P_n$ the sequence of $\delta$-measures at the points $1/n$ on the real line). However, it is "yes" if the set $A$ is open.

PS I think giving the right answer should be a reasonable prerequisite for declaring a question to be too easy for "research level".

Portmanteau theorem. $\endgroup$