Weak convergence of Monge-Ampère measures on compact Hermitian manifolds

We give a sufficient condition on a sequence of uniformly bounded $\omega$-plurisubharmonic functions, $\omega$ being a Hermitian metric, for which the sequence of associated Monge-Ampère measures converges weakly. This criterion can be used to obtained a bounded $\omega$-plurisubharmonic solution to the Monge-Ampère equation.