PomeCorollary 16.8.14.FE600747C81E Let $\varphi ,\psi \in \mathrm{QPSH}(X)$. Assume that $\varphi \preceq _ P \psi $. Then $\varphi \preceq _{\mathcal{I}}\psi $.
Corollary 16.8.14FE600747C81E
Proof. Thanks to 16.8.13, we have $\varphi \lor \psi \sim _ P \psi $. Take a closed smooth real $(1,1)$-form $\theta $ on $X$ such that $\varphi ,\psi \in \mathrm{PSH}(X,\theta )_{>0}$. Then by proposition 16.6.18, we have
\[ \psi \sim _{\mathcal{I}} P_{\theta }[\psi ]= P_{\theta }\left[\varphi \lor \psi \right]\sim _{\mathcal{I}} \varphi \lor \psi . \]
Our assertion follows from the $\mathcal{I}$-version of 16.8.13. $\square$
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