Let $(\varphi_{j})\in PSH(X,\omega)^{\mathbb{N}}$.

1) If $(\varphi_{j})$ is uniformly bounded from above on $X$, then either $\varphi_{j}$ converges uniformly to $-\infty$ on $X$ or the sequence $(\varphi_{j})$ is relatively compact in $L^{1}(X)$.

2) If $\varphi_{j}\rightarrow\varphi$ in $L^{1}(X)$, then $\varphi$ coincides almost everywhere with a unique function $\varphi^{*}\in PSH(X,\omega)$. Moreover

3) In particular if $\varphi_{j}$ is decreasing, then either $\varphi_{j}\rightarrow-\infty$ or $\varphi=\lim\varphi_{j}\in PSH(X,\omega)$. Similarly, if $\varphi_{j}$ is increasing and uniformly bounded from above then $\varphi:=(\lim\varphi_{j})^{*}\in PSH(X,\omega)$, where ${\cdot}^{*}$ denotes the upper-semi-continuous regularization.