MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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

share|cite|improve this question
For more information about the relation between weak and a kind of "set-wise" convergence look at the so-called Portmanteau theorem. – Jochen Wengenroth Mar 19 '13 at 7:09
up vote 1 down vote accepted

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".

share|cite|improve this answer
I stand corrected. – Nik Weaver Mar 19 '13 at 4:55

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.