Continuous solutions to Monge-Ampère equations onHermitian manifolds for measures dominated by capacity

We prove the existence of a continuous quasi-plurisubharmonic solution to the Monge-Ampère equation on a compact Hermitian manifold for a very general measure on the right hand side. We admit measures dominated by capacity in a certain manner, in particular, moderate measures studied by Dinh-Nguyen-Sibony. As a consequence, we give a characterization of measures admitting Hölder continuous quasi-plurisubharmonic potential, inspired by the work of Dinh-Nguyen.