Domains of definition of Monge-Amp\`ere operators on compact K\"ahler manifolds

Let $(X,\omega )$ be a compact K"ahler manifold. We introduce and study the largest set $DMA(X,\omega )$ of $\omega$-plurisubharmonic (psh) functions on which the complex Monge-Amp`ere operator is well defined. It is much larger than the corresponding local domain of definition, though still a proper subset of the set $PSH(X,\text{om})$ of all $\text{om}$-psh functions. We prove that certain twisted Monge-Amp`ere operators are well defined for all $\omega$-psh functions. As a consequence, any $\text{om}$-psh function with slightly attenuated singularities has finite weighted Monge-Amp`ere energy.