I have run into the following statement in the literature (e.g. here, p.5, after Theorem 1.1): that weak convergence of positive $(1,1)$-currents on a complex manifold is equivalent to $L^1$ (I presume $L^1_{loc}$ in the non-compact case) convergence of their $dd^c$-potentials which are normialized (I presume, having a fixed mean). I tried to search the book of Demailly for it but couldn't quite find such a statement. What is the precise statement and how can it be proved?