. 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 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

$\displaystyle\int\limits_{V}\Omega\leq e^{\alpha}Ce^{-\alpha/\operatorname{Cap}_{\gamma}(V)^{1/n}}\,,$

for all open sets $V\subset X$.