Corollary 5.25.30AD144F69807

Proof. On each connected coordinate ball, after a common smooth correction, the limit of a decreasing sequence of plurisubharmonic functions is either plurisubharmonic or identically $-\infty $. Fatou's lemma, applied to the negative parts, excludes the second alternative because of the $L^1$ bound. Hence $u$ is quasi-plurisubharmonic, and the inequality $\theta +\mathrm{dd}^{\mathrm{c}}u\geq 0$ passes to the decreasing limit. Finally, the nonnegative functions $u_1-u_ j$ increase to $u_1-u$, so monotone convergence gives $u_ j\to u$ in $L^1$. $\square$