  We consider degenerate Monge-Amp\`ere equations on compact Hessian manifolds.
We establish compactness properties of the set of normalized quasi-convex
functions and show local and global comparison principles for twisted
Monge-Amp\`ere operators. We then use the Perron method to solve Monge-Amp\`ere
equations whose RHS involves an arbitrary probability measure, generalizing
works of Cheng-Yau, Delano\"e, Caffarelli-Viaclovsky and Hultgren-\"Onnheim.
The intrinsic approach we develop should be useful in deriving similar results
on mildly singular Hessian varieties, in line with the Strominger-Yau-Zaslow
conjecture.
