Remarks on weak convergence of complex Monge–Ampère measures

Abstract

Let $\left(u_j\right)$ be a decreasing sequence of psh functions in the domain of definition $D$ of the Monge–Ampère operator on a domain $\Omega$ of ${ℂ}^n$ such that $u={inf}_j{u}_j$ is plurisubharmonic on $\Omega$. In this paper we are interested in the problem of finding conditions insuring that $\underset{j\to +\infty }{lim}\int \phi {\left(d{d}^c{u}_j\right)}^n=\int \phi NP{\left(d{d}^{c}u\right)}^n$for any continuous function on $\Omega$ with compact support, where $NP{\left(d{d}^{c}u\right)}^n$ is the nonpolar part of ${\left(d{d}^{c}u\right)}^n$, and conditions implying that $u\in D$. For $u_j=max(u,-j)$ these conditions imply also that $\underset{j\to +\infty }{lim}{\int }_K{\left(d{d}^c{u}_j\right)}^n={\int }_{K}NP{\left(d{d}^{c}u\right)}^n$for any compact set $K\subset \{u>-\infty \}$.