Lemma 17.15.19.5DB0F974F922 Let $\Gamma ,\Gamma '\in \mathrm{TC}(X,\theta )_{>0}$ and $\omega $ be a closed smooth positive $(1,1)$-form on $X$. Then
\[ P_{\theta +\omega }[\Gamma \lor \Gamma ']=P_{\theta +\omega }[\Gamma ]\lor P_{\theta +\omega }[\Gamma ']. \]
Proof.
We first observe that
\[ \left(P_{\theta +\omega }[\Gamma \lor \Gamma ']\right)_{\max }=\left( P_{\theta +\omega }[\Gamma ]\lor P_{\theta +\omega }[\Gamma ']\right)_{\max }=\Gamma _{\max }\lor \Gamma '_{\max }. \]
Let $\tau \in \mathbb {R}$ be less than this common value. We need to show that
\[ (\Gamma \lor \Gamma ')_{\tau }\sim _ P \left(P_{\theta +\omega }[\Gamma ]\lor P_{\theta +\omega }[\Gamma '] \right)_{\tau }. \]
We need the formula (17.15.16) proved in the proof of 17.15.14:
\[ \begin{split} (\Gamma \lor \Gamma ')_{\tau }\sim _ P\operatorname *{sup}_{}{}\! {}^{*}\left\{ \lambda \Gamma _{\rho }+(1-\lambda )\Gamma '_{\rho '}: \lambda \in (0,1),\rho ,\rho '\in \mathbb {R},\lambda \rho +(1-\lambda )\rho '=\tau \right\} . \end{split} \]
A similar result holds with $P_{\theta +\omega }[\Gamma ]$ and $P_{\theta +\omega }[\Gamma ']$ in place of $\Gamma $ and $\Gamma '$. So our assertion is a direct consequence of proposition 16.8.27 and proposition 16.8.28.
$\square$
Pome
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