|\phi(x)|,|\phi(y)|\}\ 
d(x,y) + \max \{ |v(x)|,|v(y)| \}\ \|\phi\|_{\hbox{Lip}}\ d(x,y)\\
  &\leq& \big( g_v(x)|\phi(x)|+g_v(y)|\phi(y)| \big)\ d(x,y) + (|v(x)|+|v(y)|)\ \|\phi\|_{\hbox{Lip}}\ d(x,y) \\
& = & d(x,y)\ ( g_{v \phi}(x) + g_{v \phi}(y) ),\end{aligned}$$ with $g_{v \phi} := g_v |\phi| + |v|\|\phi\|_{\hbox{Lip}} .$

Finally, before stating Theorem \[theorem:Lions\], we recall the following Lemma attributed to H. Brézis and E. Lieb:

\[BL\] Let $p \in [1, \infty)$. If $f_n \rightarrow f$ weakly in $L^{p}(X,\mu)$ and $f_n \rightarrow f$ $\mu$-almost everywhere, then $$\lim_{n \rightarrow \infty} \left(  \int_{X}  |f_n|^{p} d\mu - \int_{X}  |f_n - f|^{p} d \mu \right) = \int_{X}  |f|^{p} d \mu.$$

With those results at hand, we have:

\[theorem:Lions\] If $(X, d, \mu)$ is an Ahlfors lower $s$-regular compact metric-measure space, and $1 < p < s$, then for every sequence $\left\{ u_n \right\}$ in $M^p_1(X,\mu)$ such that $u_n \rightarrow u$ weakly in $M^p_1(X,\mu)$ and $u_n \rightarrow u$ strongly in $L^p(X,\mu)$, there exists a subsequence $\left\{ u_n \right\}$ and a countable set $I$ such that $$\label{eq:form}
\mu \llcorner |u_n|^{p^*} \to \mu \llcorner |u|^{p^*}  + \sum_{i \in I} \mu_i \delta_{x_i}$$ in $\mathcal{M} (X)$, where $x_i \in X$ for every $i \in I$.

We begin with two observations:

1.  Let $v_n:= u_n - u$, and fix some 