Convergence in Capacity of the Perron-Bremermann Envelope

摘要

In [CK1], Cegrell and Kolodziej constructed a sequence of measures in a ball μ_j converging to μ in the weak~* topology such that the solutions of the Dirichlet problems (dd~cu_j)~n = dμ_j, u_j = 0 on the boundary are uniformly bounded yet u_j does not converge to u, the solution of the Dirichlet problem (dd~cu)~n = dμ, u = 0 on the boundary. In [CK2] the authors gave conditions on the Monge—Ampere mass of the solutions u_j, with fixed continuous boundary values φ, that guarantee the stability of the complex Monge-Ampere operator. They introduced the set A(μ) of all solutions u of the Dirichlet problem u ∈ F(φ), (dd~cu)~n = gdμ, where μ is a positive finite measure that does not put mass on pluripolar sets and where g varies over all μ-measurable functions satisfying 0 ≤ g ≤ 1. Cegrell and Kolodziej proved that, in A(μ), weak~* convergence is equivalent to convergence in capacity.