  Let $X$ be a compact K\"ahler manifold and $\om$ a smooth closed form of
bidegree $(1,1)$ which is nonnegative and big. We study the classes ${\mathcal
E}_{\chi}(X,\om)$ of $\om$-plurisubharmonic functions of finite weighted
Monge-Amp\`ere energy. When the weight $\chi$ has fast growth at infinity, the
corresponding functions are close to be bounded.
  We show that if a positive Radon measure is suitably dominated by the
Monge-Amp\`ere capacity, then it belongs to the range of the Monge-Amp\`ere
operator on some class ${\mathcal E}_{\chi}(X,\om)$. This is done by
establishing a priori estimates on the capacity of sublevel sets of the
solutions.
  Our result extends U.Cegrell's and S.Kolodziej's results and puts them into a
unifying frame. It also gives a simple proof of S.T.Yau's celebrated a priori
${\mathcal C}^0$-estimate.
