Monge-Amp\`ere measures of bounded potentials

Let $X$ be a compact K\"ahler manifold and let $\mu$ be a non-pluripolar measure on $X$. We give a necessary and sufficient condition for $\mu$ so that the complex Monge-Amp\`ere equation (in a K\"ahler class in $X$) having $\mu$ as the right-hand side admits a bounded solution. This implies in particular a well-known conjecture of Kolodziej that a complex Monge-Amp\`ere equation on $X$ admits a bounded solution if it has a bounded subsolution.