I was trying to prove the fact that the Lévy metric, given by

$ d_L(\mu,\nu):=\inf\lbrace\varepsilon>0\,;\, \mu((-\infty,x-\varepsilon])-\varepsilon \leq \nu((-\infty,x]) \leq \mu((-\infty,x+\varepsilon])+\varepsilon\rbrace$

for two probability measures $\mu$ and $\nu$ on $\mathbb{R}$, metrizes weak convergence. I have already proven the implication

$\mu_n\xrightarrow{\omega}\mu \,\,\Longrightarrow\,\,d_L(\mu_n,\mu)\rightarrow{}0$

The other implication turned out to be a bit tricky. I know by definition that

$\lim\limits_{n\rightarrow}\mu_n((-\infty,x]) = \mu((-\infty,x])$ for all $x\in\mathbb{R}$ and that the set containing all $(-\infty,x]$ is a generating system of the Borel $\sigma$-algebra on $\mathbb{R}.$

I'm having trouble deducing from this that any of the portmanteau conditions hold. I'd appreciate any help. Thanks a lot.


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