For any sequence $(\s_{\varepsilon},\phi_{\varepsilon})\subset\M({\Omega},{\mathbf{R}}^2)\times L^1({\Omega})$ such that $\s_{\varepsilon}\rwstar\s$ and $\phi_{\varepsilon}{\rightarrow}\phi$ in the $L^1({\Omega})$ topology, with $(\s,\phi)\in\M({\Omega},{\mathbf{R}}^2)\times L^1({\Omega})$ $$\liminf_{k{\rightarrow}+{\infty}}{\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon})\geq {\mathcal{E}}_\a(\s,\phi).$$

In this statement and throughout the paper, we make a small abuse of language by noting $(a_{\varepsilon})_{{\varepsilon}\in(0,1]}$ and calling sequence a family $\{a_{\varepsilon}\}$ labeled by a continuous parameter ${\varepsilon}\in(0,1]$. In the same spirit, we call subsequence of $(a_{\varepsilon})$, any sequence $(a_{{\varepsilon}_j})$ with ${\varepsilon}_j{\rightarrow}0$ as $j{\rightarrow}+{\infty}$.\
To complete the $\Gamma$-convergence analysis, we establish the matching upper bound.

\[teo:sigma\_limsup\] For any $(\s,\phi)\subset\M({\Omega},{\mathbf{R}}^2)\times L^1({\Omega})$ there exists a sequence $(\s_{\varepsilon},\phi_{\varepsilon})$ such that $\s_{\varepsilon}\rwstar\s$ and $\phi_{\varepsilon}{\rightarrow}\phi$ in the $L^1({\Omega})$ topology and $$\limsup_{k{\rightarrow}+{\infty}}{\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon})\leq {\mathcal{E}}_\a(\s,\phi).$$

Moreover, under the assumption $\a>0$ we prove the equicoercivity of the sequence $({\mathcal{F}}_{\varepsilon})$.

\[teo:sigma\_equicoercive\] Assume $\a>0$. For any sequence $(\s_{\varepsilon},\phi_{\varepsilon})_{{\varepsilon}\in(0,1]}\subset\M({\Omega},{\mathbf{R}}^2)\times L^1({\Omega})$ with uniformly bounded energies, [*i.e.*]{} $$\sup_{{\varepsilon}} {\mathcal{F}}_{\varepsilon}(\s_{\varepsilon},\phi_{\varepsilon})\ <\ +{\infty},$$ there exist a subsequence ${\varepsilon}_j{\downarrow}0$ and a measure $\s\in\M_S({\overline}{\Omega},{\mathbf{R}}^2)$ such that $\s_{{\varepsilon}_j}\rightarrow \s$ with respect to the weak-$*$ convergence of measures and $\phi_{{\varepsilon}_j} \rightarrow 1$ in $L^1({\Omega})$. Moreover, $\s$ is a rectifiable measure ([*i.e.*]{} it is of the form $\s=U(M,\t,\xi)$).

Observe that letting $\a=0$ in equation  we obtain ${\mathcal{E}}_0(\sigma)=\H^1(\{x\in M\,:\, \t(x)>0\})$ where $\s=U(M,\t,\xi)$. This is exactly the functional associated with the Steiner Minimal Tree problem. Unfortunately, the hypothesis $\a>0$ is necessary in the 