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Suppose that a sequence $(\mu_{k})_{k=1}^{\infty}$ converges weakly to some measure $\mu$, where $(X,d)$ is a polish space and the measures are Borel probability measures.

If for some Borel set $A$ we have $\mu_{k}(A)=c$ (where c is a constant) for all $k$, do we know anything about $\mu(A)$? I know $\mu_{k}(A)\to \mu(A)$ if $\mu(\partial A)=0$, but I would be interested to know if there's anything we can do in this case.

Thanks in advance.

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If $A$ is open then $\mu(A) \le c$ while if $A$ is closed then $\mu(A) \ge c$. But for general Borel $A$, $\mu(A)$ could be anything. –  Nate Eldredge Feb 26 '12 at 23:13

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up vote 5 down vote accepted

No: $A=(0,2)$ and $\mu_n=\delta_{1/n}$ gives $\mu=\delta_0$ hence $\mu_n(A)=1$ for every $n\geqslant1$ but $\mu(A)=0$, while $A=[0,1)$ and $\mu_n=\delta_{-1/n}$ gives $\mu=\delta_0$ hence $\mu_n(A)=0$ for every $n\geqslant1$ but $\mu(A)=1$.

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Alright, thanks for the fast reply. –  Thomas E. Feb 26 '12 at 23:11

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