Remarks on weak convergence of complex Monge-Ampère measures

Let $(u_j)$ be a deaceasing sequence of psh functions in the domain of definition $\mathcal{D}$ of the Monge-Ampère operator on a domain $\mathrm{\Omega }$ of ${\mathbb{C}}^n$ such that $u={\inf }_j{u}_j$ is plurisubharmonic on $\mathrm{\Omega }$. In this paper we are interested in the problem of finding conditions insuring that \begin{equation*} \lim_{j\to +\infty} \int\varphi (dd^cu_j)^n=\int\varphi {\rm NP}(dd^cu)^n \end{equation*} for any continuous function on $\mathrm{\Omega }$ with compact support, where $\mathrm{N}\mathrm{P}(d{d}^{c}u{)}^n$ is the nonpolar part of $(d{d}^{c}u{)}^n$, and conditions implying that $u\in \mathcal{D}$. For $u_j=\max (u,-j)$ these conditions imply also that \begin{equation*} \lim_{j\to +\infty} \int_K(dd^cu_j)^n=\int_K {\rm NP}(dd^cu)^n \end{equation*} for any compact set $K\subset \{u>-\mathrm{\infty }\}$.