The family

$${\mathcal{F}}_{0}:=\{\varphi\in PSH(X,\omega)\,/\,\sup_{X}\varphi=0\}$$

is a compact subset of $PSH(X,\omega)$.

If $\mu$ is a probability measure such that $PSH(X,\omega)\subset L^{1}(\mu)$ then

$${\mathcal{F}}_{\mu}:=\{\varphi\in PSH(X,\omega)\,/\,\int_{X}\varphi d\mu=0\}$$

is a relatively compact subset of $PSH(X,\omega)$. In particular there exists $C_{\mu}$ such that $\forall\varphi\in PSH(X,\omega)$,

$$-C_{\mu}+\sup_{X}\varphi\leq\int_{X}\varphi d\mu\leq\sup_{X}\varphi.$$