Weak convergence of complex Monge-Amp\`ere operators on compact Hermitian manifolds

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-Amp\`ere measures associated to a sequence of $\beta$-plurisubharmonic functions. Second, this criterion is utilized to solve a degenerate complex Monge-Amp\`ere equation with an $L^1$-density. Finally, an $L^\infty$-estimate of the solution to the complex Monge-Amp\`ere equation for a finite positive Radon measure is given.