PomeProposition 16.6.18.8AE79B76D6D2 Let $\varphi \in \mathrm{PSH}(X,\theta )$. Then
\[ P_{\theta }[\varphi ]\leq P_{\theta }[\varphi ]_{\mathcal{I}},\quad \varphi \sim _{\mathcal{I}}P_{\theta }[\varphi ]. \]
Proposition 16.6.188AE79B76D6D2
Proof. It suffices to show that $\varphi \sim _{\mathcal{I}} P_{\theta }[\varphi ]$. Namely, for each $k\in \mathbb {Z}_{>0}$, we have
EA49AC910A56
16.6.20
\begin{equation} \mathcal{I}(k\varphi )=\mathcal{I}\Bigl (kP_{\theta }[\varphi ]\Bigr ). \end{equation}
Fix $k$ for now. It follows from (16.3.7) and the strong openness theorem theorem 5.16.21 that
\[ \mathcal{I}\Bigl (kP_{\theta }[\varphi ]\Bigr )=\mathcal{I}\Bigl ((k\varphi +C)\land kV_{\theta }\Bigr ), \]
when $C$ is large enough. Since $(k\varphi +C)\land kV_{\theta }\sim k\varphi $, we have
\[ \mathcal{I}\Bigl ((k\varphi +C)\land kV_{\theta }\Bigr )=\mathcal{I}(k\varphi ) \]
and (16.6.20) follows. $\square$
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