Let \((X,\omega )\) be a compact Hermitian manifold and let \(\{\beta \}\in H^{1,1}(X,{\mathbb {R}})\) be a real (1, 1)-class with a smooth representative \(\beta \) , such that \(\int _X\beta ^n>0\) . Assume that there is a bounded \(\beta \) -plurisubharmonic function \(\rho \) on X. First, we provide a criterion for the weak convergence of non-pluripolar complex Monge-Ampère measures associated to a sequence of \(\beta \) -plurisubharmonic functions. Second, this criterion is utilized to solve a degenerate complex Monge-Ampère equation with an \(L^1\) -density. Finally, an \(L^\infty \) -estimate of the solution to the complex Monge-Ampère equation for a finite positive Radon measure is given.
