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Let (X, w) be a compact Kahler manifold. We introduce and study the largest set DMA(X, w) of w-plurisubharmonic (psh) functions on which the complex Monge-Ampere 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, w) of all w-psh functions. We prove that certain twisted Monge-Ampere operators are well defined for all w-psh functions. As a consequence, any w-psh function with slightly attenuated singularities has finite weighted Monge-Ampere energy.

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