Approximation of maximal plurisubharmonic functions

Let $u$ be a maximal plurisubharmonic function in a domain $\mathrm{\Omega }\subset {\mathbb{C}}^n$ ($n\ge 2$). It is classical that, for any $U⋐\mathrm{\Omega }$, there exists a sequence of bounded plurisubharmonic functions $PSH(U)\ni u_j\searrow u$ satisfying the property: $(d{d}^c{u}_j{)}^n$ is weakly convergent to $0$ as $j\to \mathrm{\infty }$. In general, this property does not hold for arbitrary sequence. In this paper, we show that for any sequence of bounded plurisubharmonic functions $PSH(U)\ni u_j\searrow u$, $(|u_j|+1)^{-a} (dd^cu_j)^n$isweaklyconvergentto$0$as$j\rightarrow\infty$,where$a>n-1$. We also generalize some well-known results about approximation of maximal plurisubharmonic functions.