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I am trying to find a sequence of probability measures $(\mu_n)_{n\in\mathbb{N}}$ which converges in the weak* topology to $\mu$, on a measurable metric space $(\Omega,\mathcal{F})$ and a closed set $F\in \mathcal{F}$ for which $\lim \sup \mu_n(F)<\mu(F)$. Can someone help me?

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  • $\begingroup$ A good candidate for $\mu$ would be a point mass. $\endgroup$ – Daniel Fischer Jan 16 '17 at 17:56

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