An example $\limsup \mu_n(F)<\mu(F)$

Question

Give examples of a sequence of measures $(\mu)_n\in\mathbb{N}$ which converges on weak topology for $\mu$, on a measurable topological space $\Omega,\mathscr{F}$ and a closed set $F\in\mathscr{F}$ which $\limsup \mu_n(F)<\mu(F)$

I do not know if the following example is right:$\delta_{-\frac{1}{n}}([−1,0])$

$\lim_{n\to\infty}\delta_{-\frac{1}{n}}([−1,0])=\delta_0$

For $n>N\in\mathbb{N}$ $\delta_0[-1,0]>\delta_{-\frac{1}{n}}([−1,0])$

which implies $\limsup \delta_{-\frac{1}{n}}([−1,0])<\delta_{0}([−1,0])=0$

Question:

Is this right?

If not what could be an alternative?

Thanks in advance!

Answer 1

If I understand your question correctly, you can take the sequence $\mu_n:=\delta_{1/n}$ which converges weak$^*$-ly to $\mu=\delta_0$ (in the topology of $C_0(\Bbb R)^*$). Let $F:=\{0\}$, then $$ \limsup_{n\to\infty} \delta_{1/n}(F) = 0 < 1 = \delta_0(F). $$