PomeLemma 16.6.28.2A3BCF353C03 Let $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Assume that $\varphi \preceq \psi $. Then
\[ P_{\theta }[\varphi ]_{\mathcal{I}}\leq P_{\theta }[\psi ]_{\mathcal{I}}. \]
Lemma 16.6.282A3BCF353C03
Proof. We may assume that $\psi \leq 0$. Observe that $P_{\theta }[\varphi ]_{\mathcal{I}}\lor \psi \sim _{\mathcal{I}} \psi $ as a consequence of proposition 5.16.8 and proposition 16.6.11. So
\[ P_{\theta }[\varphi ]_{\mathcal{I}}\lor \psi \leq P_{\theta }[\psi ]_{\mathcal{I}}. \]
The desired inequality follows. $\square$
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.