Remarks on weak convergence of complex Monge–Ampère measures

Abstract

Let (uj) be a decreasing sequence of psh functions in the domain of definition D of the Monge–Ampère operator on a domain Ω of ℂn such that u=infjuj is plurisubharmonic on Ω. In this paper we are interested in the problem of finding conditions insuring that limj→+∞∫φ(ddcuj)n=∫φNP(ddcu)nfor any continuous function on Ω with compact support, where NP(ddcu)n is the nonpolar part of (ddcu)n, and conditions implying that u∈D. For uj=max(u,−j) these conditions imply also that limj→+∞∫K(ddcuj)n=∫KNP(ddcu)nfor any compact set K⊂{u>−∞}.