Let $\Omega$ be a semi-Kähler current with $L_{loc}^{\infty}$ potentials on $V$. The Monge-Ampère measure $\Omega_{reg}^{n}$ is well defined on $V_{reg}$ and satisfies $\int_{U^{reg}}\Omega_{reg}^{n}<\infty,\text{ for all relatively compact subset }U\subset V.$

For any resolution $\pi:X\to V$, the Monge-Ampère measure $(\pi^{*}\Omega)^{n}$ is well defined on $X$ and satisfies $\pi_{*}(\pi^{*}\Omega)^{n}=j_{*}\Omega_{reg}^{n}$. Moreover if $\bar{\pi}:\bar{X}\to V$ is a resolution dominating $\pi$ (i.e. $\bar{\pi}=\pi\circ\psi$ for some bimeromorphic proper holomorphic map $\psi:\bar{X}^{\prime}\to X^{\prime}$), then $\psi_{*}(\bar{\pi}^{*}\Omega)^{n}=(\pi^{*}\Omega)^{n}$.