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Proof.
Fix u ∈ P S H ( X , ω ) u\in PSH(X,\omega) 0 ≤ u ≤ 1 0\leq u\leq 1 δ > 0 \delta>0 t = δ / ( 1 + δ ) t=\delta/(1+\delta) 0 ≤ t ≤ 1 0\leq t\leq 1
{ φ − ψ < − s − t } ⊂ { φ < ψ + δ u 1 + δ − s − t } ⊂ { φ − ψ < − s − t ψ } . \{\varphi-\psi<-s-t\}\subset\{\varphi<\frac{\psi+\delta u}{1+\delta}-s-t\}\subset\{\varphi-\psi<-s-t\psi\}.
Set φ ~ := ( ψ + δ u ) / ( 1 + δ ) − s − t ∈ P S H ( X , ω ) \tilde{\varphi}:=(\psi+\delta u)/(1+\delta)-s-t\in PSH(X,\omega)
t n ∫ ( φ − ψ < − s − t ) ω u n ≤ ∫ ( φ − ψ < − s − t ) [ 1 1 + δ ω ψ + δ 1 + δ ω u ] n ≤ ∫ ( φ < φ ~ ) [ ω + d d c φ ~ ] n . t^{n}\int_{(\varphi-\psi<-s-t)}\omega_{u}^{n}\leq\int_{(\varphi-\psi<-s-t)}\left[\frac{1}{1+\delta}\omega_{\psi}+\frac{\delta}{1+\delta}\omega_{u}\right]^{n}\leq\int_{(\varphi<\tilde{\varphi})}[\omega+dd^{c}\tilde{\varphi}]^{n}.
It follows from the
comparison principle in class ℰ 1 ( X , ω ) {\mathcal{E}}^{1}(X,\omega)
∫ ( φ < φ ~ ) [ ω + d d c φ ~ ] n ≤ ∫ ( φ < φ ~ ) [ ω + d d c φ ] n ≤ ∫ ( φ − ψ < − s − t ψ ) [ ω + d d c φ ] n . \int_{(\varphi<\tilde{\varphi})}[\omega+dd^{c}\tilde{\varphi}]^{n}\leq\int_{(\varphi<\tilde{\varphi})}[\omega+dd^{c}\varphi]^{n}\leq\int_{(\varphi-\psi<-s-t\psi)}[\omega+dd^{c}\varphi]^{n}.
Taking the supremum over all u’s yields the desired result.
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