be a smooth real $(1,1)$-form, and denote by $\alpha[\gamma]$ the set of closed almost positive $(1,1)$-currents $T$ lying in $\alpha$ with $T\geq\gamma$. It is a (weakly) compact and convex subset of the space of $(1,1)$-currents. We endow it with the pre-order relation $\preceq$ defined above. For any family $T_j$, $j\in J$ of elements of $\alpha[\gamma]$, we claim that there exists an infimum $T=\inf_{j\in J}T_j$ in $(\alpha[\gamma],\preceq)$, which is therefore unique up to equivalence of singularities. The proof is pretty straightforward: fix a smooth form $\theta$ in $\alpha$, and write $T_j=\theta+dd^c\varphi_j$ for some quasi pluri-subharmonic functions $\varphi_j$. Since $X$ is compact, $\varphi_j$ is bounded from above; therefore, upon changing $\varphi_j$ into $\varphi_j-C_j$, we may assume that $\varphi_j\leq 0$ for all $ j\in J$. We then take $\varphi$ to be the upper semi-continuous upper enveloppe of the $\varphi_j$’s, $j\in J$, and set $T:=\theta+dd^c\varphi$. It is immediate to check that $T\preceq T_j$ for all $ j$, and that for every $S\in\alpha[\gamma]$, $S\preceq T_j$ for all $ j$ implies that $S\preceq T$. We should maybe explain why $T\geq\gamma$: locally, we can choose coordinates $z=(z_1,...,z_n)$ and a form $q(z)=\sum\lambda_j|z_j|^2$ such that $dd^cq\leq\gamma$ and $dd^cq$ is arbitrarily close to $\gamma$. Writing $\theta=dd^c\psi$ for some smooth local 