. Let $X$ be a connected compact complex manifold of complex dimension $n$, let $\gamma$ be a big closed positive $(1,1)$-current with bounded local potentials.
(A). If $X$ is Kähler and $\gamma$ possesses continuous local potentials then there exist a constant $C=C(\gamma)>0$ such that $\operatorname{Cap}_{\gamma}(\{\psi<-t\})\leq C/t$ for all $\psi\in{\cal P}^{0}_{\gamma}$ and $t>0$. Moreover the constant $C$ stay bounded for pertutbations of $\gamma$ satisfying the hypothesis $(C1)$ and $(C2a)$ of the statement $(C)$ in theorem 3.
(B). If $\gamma^{n}/\Omega\in L\log L(X)$, for a smooth volume form $\Omega>0$ then the conclusion of statement $(A)$ hold whith a constant $C=C(\gamma,\Omega)>0$ which stay bounded for pertutbations of $\gamma$ satisfying the hypothesis $(C1)$ and $(C2b)$ of the statement $(C)$ in theorem 3.