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 $\mathcal{E}(X,\theta ,\phi )$, for a model potential $\phi$. 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 ${\mathcal{E}}^{\frac{n}{n-1}}(X,\theta ,\phi )$ (provided $n>1$), while they have the same singularities of $\phi$ when $n=1$.