Let $L\rightarrow X$ be a positive holomorphic line bundle equipped with a smooth strictly positive metric $h$, and set $\omega:=\Theta_{h}>0$.

Then for every $\varphi\in PSH(X,\omega)$, there exists a sequence $\varphi_{j}\in PSH(X,\omega)\cap{\mathcal{C}}^{\infty}(X)$ such that $\varphi_{j}$ decreases towards $\varphi$.