that $x_n\to x$ and, for any $R>0$, $$\limsup_{n\to +\infty}\int_{B_R}\textbf{f}_n\left(x_n+\frac{\lambda}{\sqrt{n}},\theta_\lambda(\nu(n),E(n),g(n))\right)d\lambda <+\infty,$$ then the integrand is bounded for a.e. $\lambda$. By assumption on $\textbf{f}_n$, for any $n$, $$\theta_\lambda(\nu(n),E(n),g(n))=\theta_{x_n\sqrt{n}+\lambda}({\nu}_n',{E}_n,{g}_n),$$ hence it follows that $$(\nu(n),E(n),g(n))=\theta_{x_n\sqrt{n}}({\nu}_n',{E}_n,{g}_n).$$ For any $R>0$, there exists $C_R>0$ such that for any $n>0$, noting $B_R(x)$ the closed ball of radius $R$ centred at a point $x$, $$\begin{aligned}
\int_{B_R}\textbf{f}_n\left(x_n+\frac{\lambda}{\sqrt{n}},\theta_\lambda(\nu_n,E_n,g_n)\right)d\lambda &=\int_{B_R}\frac{1}{\pi}\int_{{\mathbb R}^2}\frac{\chi(y)}{m_V\left(x_n+\frac{\lambda}{\sqrt{n}}\right)}d(\theta_{\lambda+x_n\sqrt{n}}\sharp {g}_n(y))d\lambda\\
&=\frac{1}{\pi}\int_{B_R}\int_{{\mathbb R}^2}\frac{\chi(y-x_n\sqrt{n}-\lambda)}{m_V\left(x_n+\frac{\lambda}{\sqrt{n}}\right)}d{g}_n(y)d\lambda\\
&=\frac{1}{\pi}\int_{{\mathbb R}^2}\chi\ast\left(\indic_{B_R(x_n\sqrt{n})}\frac{1}{m_V(./\sqrt{n})}  \right)(y)d{g}_n(y)<C_R.\end{aligned}$$ This, inequalities and the fact that ${g}_n$ is bounded below imply that ${g}_n(B_R(x_n\sqrt{n}))$ is bounded independently of $n$. Hence by the same argument as in [@2DSandier Lemma 4.4], we have the convergence of a subsequence of $(\nu_n,E_n,g_n)$.\
2) We have the $\Gamma$-liminf property: if $(x_n,\nu_n,E_n,g_n)\to(x,\nu,E,g)$ as $n\to +\infty$, then, by Fatou’s Lemma, $$\liminf_{n\to +\infty} \mathbf{f}_n(x_n,\nu_n,E_n,g_n)\geq f(x,\nu,E,g):=\frac{1}{\pi}\int\frac{\chi(y)}{m_V(x)} dg(y),$$ obviously if the left-hand side is finite. Therefore, Theorem \[erg\] applies and implies that:\

1.  The measure $Q_n$ admits a subsequence which converges to a measure $Q$ which has $\mu_V$ as first marginal.

2.  It holds that $Q$-almost every $\displaystyle (x,\nu,E,g)$ is of the form $\displaystyle \lim_{n\to +\infty}(x_n, \theta_{x_n\sqrt{n}}({\nu}'_n,{E}_n,{g}_n))$.

3.  The measure $Q$ is $T_{\lambda(x)}$-invariant.

4.  We have $\displaystyle \liminf_{n\to +\infty} \mathbf{F}_n({\nu}'_n,{E}_n,{g}_n)\geq \frac{1}{\pi} \int_{{\mathbb R}^2}\left(\int_{{\mathbb R}^2}\frac{\chi(y)}{m_V(x)} dg(y)\right) dQ(x,\nu,E,g)$.

5.  $\displaystyle \frac{1}{\pi}\int \int \frac{\chi(y)}{m_V(x)}dg(y) dQ(x,\nu,E,g)=\int \left( \lim_{R\to +\infty}\dashint_{B_R} \int 