I'm reading about convergence vague. As a comment it says the following:
$\mu_{n}\overset{\nu}{\rightarrow}\mu$ implies $\mu(\mathbb{R})\leq\liminf_{n}\mu_{n}(\mathbb{R}).$
I'm trying to prove this but I don't get it.
The definition given is: Let $\mu_{n}$ and $\mu$ be finite measures on $(\mathbb{R},\mathcal{B}_{\mathbb{R}}).$ If $\mu_{n}(a,b]\rightarrow\mu(a,b]$ for every finite interval which $\mu(\{a\})=\mu(\{b\})=0,$ then $\mu_{n}$ converges vaguely to $\mu.$
Any kind of help is thanked in advanced.