  Let $X$ be a compact complex manifold which admits a hermitian metric
satisfying a curvature condition introduced by Guan-Li. Given a semipositive
form $\theta$ with positive volume, we define the Monge-Amp\`ere operator for
unbounded $\theta$-psh functions and prove that it is continuous with respect
to convergence in capacity. We then develop pluripotential tools to study
degenerate complex Monge-Amp\`ere equations in this context, extending recent
results of Tosatti-Weinkove, Kolodziej-Nguyen, Guedj-Lu and many others who
treat bounded solutions.
