Monge-Ampere type equations on compact Hermitian manifolds with bounded mass property

In this paper, we study possibly non-closed big (1, 1)-forms on a compact Hermitian manifold satisfying the bounded mass property. We propose several criteria for the existence of rooftop envelopes. As applications, we establish the existence of solutions to complex Monge-Ampere type equations with prescribed singularities, allowing for non-pluripolar measures on the right-hand side. We also obtain stability results when singularity types vary, by extending the Darvas-Di Nezza-Lu distance to the Hermitian context.