. Let $X$ be a connected compact complex manifold of complex dimension $n$, let $\gamma$ be a closed real $(1,1)$-current with continuous local potentials or a closed positive $(1,1)$-current with bounded local potentials and let $\Omega>0$ be a smooth volume form. Then there exist constants $\alpha=\alpha(\gamma,\Omega)>0$, $C=C(\gamma,\Omega)>0$ such that $\int_{X}-\psi\,\Omega\leq C$ and $\int_{X}e^{-\alpha\psi}\,\Omega\leq C$ for all $\psi\in{\cal P}^{0}_{\gamma}$.