PomeProposition 22.1.36.74CC0BF97B41 Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )$ satisfying (22.1.35). Assume that $\omega $ is a closed smooth positive $(1,1)$-form on $X$. Then
\[ P_{\theta +\omega }\left[ \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right]_{\mathcal{I}}= \operatorname *{sup}_{i\in I}{}\! {}^{*} P_{\theta +\omega }\left[ \Gamma ^{i} \right]_{\mathcal{I}}. \]
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