Uniqueness in E(X,ω)

We prove uniqueness for the Dirichlet problem for the complex Monge–Ampère equation on compact Kähler manifolds in the case of probability measures vanishing on pluripolar sets. The proof uses the mass concentration technique due to Kołodziej coupled with inequalities for mixed Monge–Ampère measures and the comparison principle. © 2009 Elsevier Inc. All rights reserved.