Remarks on weak convergence of complex Monge-Amp\`ere measures

Let $(u_j)$ be a deaceasing sequence of psh functions in the domain of definition $\cal D$ of the Monge-Amp\`ere operator on a domain $\Omega$ of $\mathbb{C}^n$ 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 \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 $\Omega$ with compact support, where ${\rm NP}(dd^cu)^n$ is the nonpolar part of $(dd^cu)^n$, and conditions implying that $u\in \cal D$. 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 $K\subset\{u>-\infty\}$.