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Proof.
Fix u ∈ PSH ( X , ω ) u\in\PSH(X,\omega) 0 ≤ u ≤ 1 0\leq u\leq 1 ψ t := ( 1 − t ) ψ + t u \psi_{t}:=(1-t)\psi+tu
{ φ < ψ } ⊆ { φ < ψ t } ⊆ { φ < ( 1 − t ) ψ + t } . \{\varphi<\psi\}\subseteq\{\varphi<\psi_{t}\}\subseteq\{\varphi<(1-t)\psi+t\}.
since ψ ≤ 0 \psi\leq 0 MA ( ψ t ) ≥ t n MA ( u ) \MA(\psi_{t})\geq t^{n}\MA(u) 3.1
t n ∫ { φ < ψ } MA ( u ) ≤ ∫ { φ < ψ } MA ( ψ t ) ≤ ∫ { φ < ψ t } MA ( ψ t ) ≤ ∫ { φ < ψ t } MA ( φ ) ≤ ∫ { φ < ( 1 − t ) ψ + t } MA ( φ ) , t^{n}\int_{\{\varphi<\psi\}}\MA(u)\leq\int_{\{\varphi<\psi\}}\MA(\psi_{t})\leq\int_{\{\varphi<\psi_{t}\}}\MA(\psi_{t})\\
\leq\int_{\{\varphi<\psi_{t}\}}\MA(\varphi)\leq\int_{\{\varphi<(1-t)\psi+t\}}\MA(\varphi),
where the third inequality follows from the
comparison principle (6.7 u u