The General Definition of the Complex Monge-Amp\`ere Operator on Compact K\"ahler Manifolds

We introduce a wide subclass $\mathcal{F}(X,\omega )$ of quasi-plurisubharmonic functions in a compact K"ahler manifold, on which the complex Monge-Amp`ere operator is well-defined and the convergence theorem is valid. We also prove that $\mathcal{F}(X,\omega )$ is a convex cone and includes all quasi-plurisubharmonic functions which are in the Cegrell class.