\lim_{j{\rightarrow}+{\infty}} {\mathcal{E}}_{f_k}(w^i_j,{\Omega}_i)= \lim_{j{\rightarrow}+{\infty}} {\mathcal{E}}_{f}(w^i_j,{\Omega}_i)={\mathcal{E}}_{f}(u_k,{\Omega}_i),\\  
        & J_{w^j_i} \mbox{ is contained in a finite union of segments for any } j\in {\mathbf{N}},\\
        & \int_{\de {\Omega}_i}|Tw^i_j-Tu_k|\dH^1{\rightarrow}0\mbox{ where } T:BV({\Omega}_i){\rightarrow}L^1(\de {\Omega}_i) \mbox{ denotes the trace operator}.\\
     \end{aligned}$$ Let us now define globally $$w_j:=\sum_j w_j^i 1_{{\Omega}_i}.$$ From the above properties, we have $w_j\rwstar u_k$, $$\label{eq:convEfw}
\lim{\mathcal{E}}_{f}(w^i_j,\hat {\Omega})={\mathcal{E}}_{f}(u_k,{\Omega}_i)$$ and $$\label{eq:diffukwj}
|D w_j - D u_k|(\cup_i\de{\Omega}_i)\, \to\, 0\quad\mbox{as $j\to\infty$}.$$ Eventually, using a diagonal argument, we have proved the existence of a sequence $(w_j)\subset \P(\hat{\Omega})$ complying to items *(a)*,  *(b)* and *(d)* of the lemma. Moreover, item *(c)* is the consequence of  and  and item *(e)* follows from  and .

Going back to the $\H^1$-rectifiable measures $\s=U(M_\s,\t_\s,\xi_\s)$, we define the sequence $$\s_j:=-Dw^\perp_i-\g.$$ We recall that $\g=U(M_\g,\t_\g,\xi_\g)$ with $M_\g\subset \cup \de {\Omega}_i$. In particular $\g=-Du^\perp{ \!\!\mbox{{ \Large$\llcorner$}} }(\cup_i \de{\Omega}_i)$. We deduce from the previous lemma:

\[lem:density\] There exists a sequence $(\s_j)\in\M_S({\overline}{\Omega})$ with the properties:

1.  $\s_j{\rightarrow}\s$ with respect to weak-$*$ convergence of measures,

2.  $\s_j= U(M_{\s_j},\t_{\s_j},\xi_{\s_j})$ with $M_{\s_j}$ contained in a finite union of segments,

3.  $\limsup_{j{\rightarrow}{\infty}}{\mathcal{E}}_\a(\s_j,1)\leq{\mathcal{E}}_\a(\s,1)$.

Construction of a recovery sequence
-----------------------------------

Let us prove the $\Gamma$-limsup inequality stated in Theorem \[teo:sigma\_limsup\]. Recall that the latter consists in finding a sequence $(\s\e,\phi\e)$ for any given couple $(\s,\phi)\in\M(\overline{\Omega},{\mathbf{R}}^2)\times L^1({\Omega})$ 