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

We introduce a wide subclass ${\cal F}(X,\omega)$ 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 ${\cal F}(X,\omega)$ is a convex cone and includes all quasi-plurisubharmonic functions which are in the Cegrell class.