$\phi$ in $\hbox{Lip}_c(X)$. The hypotheses, Proposition \[wlozenie\] and Lemma \[prod\] when applied to $f_n := v_n \phi$ give $$\label{istot}
    \|v_n \phi\|_{L^{p^*}(X,\mu)} \leq C\left(\|v_n \phi\|_{L^{p}(X,\mu)}+\| g_{v_n} \phi\|_{L^{p}(X,\mu)} + 
    \|\phi\|_{\hbox{Lip}}  \| v_n \|_{L^{p}(X,\mu)}\right) .$$

2.  The hypotheses also imply that

    -   $\|v_n \phi\|_{L^{p}(X,\mu)} \to 0$ and $\| v_n \|_{L^{p}(X,\mu)}  \to 0$,

    -   $\mu \llcorner  |v_n|^{p^{\ast}} \to \bar{\mu}$ and $\mu \llcorner |g_{v_n}|^{p} \to \nu$ for some $\bar{\mu}$ and $\nu$ in $\mathcal{M}(X)$.

Those observations yield the reverse Hölder inequality $$\| \phi \|_{L^{p^*}(X,\bar{\mu})} \leq C \| \phi \|_{L^{p}(X,\nu)} ,$$ and Lemma \[lemma:rhol\] ensures that the mesure $\bar{\mu}$ has the form $$\label{eq:suma}
\bar{\mu} = \sum_{i \in I} \mu_i \delta_{x_i}.$$

Now use Lemma \[BL\] when $f_n=u_n \phi^{\frac{1}{p^*}}$, where $\phi$ is a non-negative function in $C_c(X)$, and $$\lim_{n \rightarrow \infty} \left(  \int_{X} \phi |u_n|^{p^{*}} d \mu - \int_{X}  \phi  |v_n|^{p^{*}} d \mu \right) =
\int_{X}  \phi  |u|^{p^{*}} d \mu$$ follows. Then recall that $\mu \llcorner |v_n|^{p^{*}} \to \bar{\mu}$ in $\mathcal{M}(X)$, to infer $$\label{eq:pos}
\lim_{n \rightarrow \infty} \int_X \phi |u_n|^{p^{*}} d \mu  =  \int_X \phi\   d \bar{\mu} + \int_X \phi |u|^{p^{*}}  d \mu.$$

Since every continuous function of compact support, say $\phi$, can be written as $\phi=\phi_{+}-\phi_{-}$, where $\phi_{+}$ and $\phi_{-}$ are non-negative with compact support, one concludes that 