Let $\psi\colon N_{\mathbb{R}}\to\mathbb{R}$ be a concave function such that $|\Psi-\psi|$ is bounded. Let $\mathcal{M}_{M}(\psi)$ be the Monge-Ampère measure associated to $\psi$ and the lattice $M$. We will denote by ${\overline{\mathcal{M}}}_{M}(\psi)$ the measure on $N_{\Sigma}$ given by

$${\overline{\mathcal{M}}}_{M}(\psi)(E)=\mathcal{M}_{M}(\psi)(E\cap N_{\mathbb{R}})$$

for any Borel subset of $N_{\Sigma}$.