Continuity of the Complex Monge-Ampère Operator on Compact Kähler Manifolds

We prove several approximation theorems of the complex Monge-Ampère operator on a compact Kähler manifold. As an application we give a simple proof of a recent result of Guedj and Zeriahi on a complete description of the range of the complex Monge-Ampère operator in E(X,ω), which is the class of ω-plurisubharmonic functions with vanishing complex Monge-Ampère mass on all pluripolar sets. We also establish the extension of Calabi’s uniqueness theorem in E(X,ω). This solves an open problem given by Guedj and Zeriahi. As a by-product we moreover obtain a stability theorem of solutions of complex Monge-Ampère equations.