Please consider the following basic result on the weak convergence of nets of radon measures, which can be found in a book of Bogachev:
I wonder whether I'm missing something, but I don't get why he claims that the integrals of $f$ and $g$ with respect to $\mu_\alpha$ differ by less than $3\varepsilon$ and the integrals of $f$ with respect to $\mu_\alpha$ and $\mu$ differ by less than $7\varepsilon$.
It's clearly not important for the result, but I've got the strange feeling that I'm missing something.
Shouldn't we have \begin{equation}\begin{split}\left|(\mu_\alpha-\mu)(f-g)\right|&\le\underbrace{\int_K\underbrace{|f-g|}_{<\:\varepsilon}\:{\rm d}|\mu_\alpha-\mu|}_{\le\:\varepsilon(\left\|\mu_\alpha\right\|+\left\|\mu\right\|)\:\le\:2\varepsilon}\\&\;\;\;\;\;\;\;\;\;\;\;\;+\underbrace{\int_{K^c}\underbrace{|f-g|}_{\le\:2}\:{\rm d}|\mu_\alpha-\mu|}_{\le\:2(|\mu_\alpha|+|\mu|)(K^c)\:<\:2\varepsilon}<4\varepsilon\end{split}\tag1\end{equation} and hence $$|(\mu_\alpha-\mu)f|\le|(\mu_\alpha-\mu)(f-g)|+|(\mu_\alpha-\mu)g|<5\varepsilon?\tag2$$
BTW, do we really need to assume that the $\mu_\alpha$ are Radon? I think this is redundant by the uniform tightness assumption. It's only important that $\{\mu\}$ is tight as well in order to ensure the existence of $K$ (which is clearly the case when $\mu$ is Radon).
