PomeProposition 16.6.5.26D170DCA6A1 Let $\varphi ,\varphi ',\psi ,\psi '\in \mathrm{QPSH}(X)$ and $\lambda >0$. Assume that $\varphi \sim _{\mathcal{I}}\psi $ and $\varphi '\sim _{\mathcal{I}}\psi '$. Then
\[ \varphi \lor \varphi '\sim _{\mathcal{I}} \psi \lor \psi ',\quad \varphi +\varphi '\sim _{\mathcal{I}} \psi +\psi ',\quad \lambda \varphi \sim _{\mathcal{I}}\lambda \psi . \]
Similarly, if $(\varphi _ i)_{i\in I}$, $(\psi _ i)_{i\in I}$ are two non-empty uniformly bounded from above families in $\mathrm{PSH}(X,\theta )$ for some closed smooth real $(1,1)$-form $\theta $ on $X$ such that $\varphi _ i\sim _{\mathcal{I}}\psi _ i$ for all $i\in I$, then
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*} \varphi _ i\sim _{\mathcal{I}}\operatorname *{sup}_{i\in I}{}\! {}^{*} \psi _ i. \]
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