Entropy for Monge-Ampère measures in the prescribed singularities setting

In this note, we generalize the notion of entropy for potentials in a relative full Monge–Amp`ere mass E(X,θ,φ), for a model potential φ. We then investigate stability properties of this condition with respect to blow-ups and perturbation of the cohomology class. We also prove a Moser–Trudinger type inequality with general weight and we show n that functions with finite entropy lie in a relative energy class E n−1 (X, θ, φ) (provided n > 1), while they have the same singularities of φ when n = 1.