Proposition 5.16.8`B9C4FE156D00`

Proposition 5.16.8.B9C4FE156D00 Let $\varphi ,\psi \in \mathrm{PSH}(X)$, $\lambda \in \mathbb {R}_{\geq 0}$ and $x\in X$. Then

\[ \begin{aligned} \nu (\varphi \lor \psi ,x)=& \min \{ \nu (\varphi ,x),\nu (\psi ,x)\} ,\\ \nu (\varphi +\psi ,x)=& \nu (\varphi ,x)+\nu (\psi ,x),\\ \nu (\lambda \varphi ,x)=& \lambda \nu (\varphi ,x). \end{aligned} \]

Proof. All properties are local, so we may assume that $X=B_ n(0,1)$ for some $n\in \mathbb {N}$. All properties follow directly from proposition 5.16.4. $\square$

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BibTeX citation

@misc{pome-tag-B9C4FE156D00,
  author = {{Pome contributors}},
  title = {Proposition 5.16.8},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/B9C4FE156D00/}
}

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Proposition 5.16.8B9C4FE156D00

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Proposition 5.16.8`B9C4FE156D00`