Let $V$ be a $\mathbb{Q}$-Gorenstein Kähler normal $n$-dimensional complex space with only log terminal singularities. Let $\Omega$ be a semi-Kähler current with $L^{\infty}_{loc}$ potential and adapted Monge-Ampère measure. Let $h$ be the singular metric on the canonical sheaf such that $\Omega^{n}=v(h)$. We define

where the equality is to be taken in the sense of currents.

The metric $\Omega$ will be called a singular Kähler-Einstein metric if $Ric(\Omega)=c\Omega$ for some $c\in\mathbb{R}$.