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-Ampère 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 that functions with finite entropy lie in a relative energy class En/n-1(X, θ, φ) (provided $n>1$), while they have the same singularities of φ when $n=1$.