  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
$K\subset\{u>-\infty\}$.
