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 $$P(B)=0$$ ???



Weak convergence in measure for negligible sets.
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".


