  We prove the existence of a continuous quasi-plurisubharmonic solution to the
Monge-Amp\`ere equation on a compact Hermitian manifold for a very general
measre 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 H\"older
continuous quasi-plurisubharmonic potential, inspired by the work of
Dinh-Nguyen.
