Proposition 16.3.2`C1971A5F3DFD`

Proposition 16.3.2.C1971A5F3DFD The operator $\land $ is a well-defined commutative, associative binary operator

\[ \bigl (\mathrm{PSH}(X,\theta )\cup \{ -\infty \} \bigr )\times \bigl (\mathrm{PSH}(X,\theta )\cup \{ -\infty \} \bigr )\rightarrow \mathrm{PSH}(X,\theta )\cup \{ -\infty \} . \]

Proof. We first show that the map is well-defined. For this purpose, take $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. When the set in (16.3.1) is empty, there is nothing to prove. So let us assume that the set is not empty.

Define

\[ \gamma =\operatorname *{sup}_{}{}\! {}^{*}\left\{ \eta \in \mathrm{PSH}(X,\theta ):\eta \leq \varphi ,\eta \leq \psi \right\} . \]

Then by proposition 5.14.2, we find that $\gamma \in \mathrm{PSH}(X,\theta )$ and hence $\gamma $ is a candidate for the supremum in (16.3.1). Therefore, $\gamma \leq \varphi \land \psi $. The reverse inequality follows from the definition of $\gamma $, so

\[ \varphi \land \psi =\gamma \in \mathrm{PSH}(X,\theta ). \]

The commutativity and the associativity of $\land $ follow directly from the defining maximality property. $\square$

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BibTeX citation

@misc{pome-tag-C1971A5F3DFD,
  author = {{Pome contributors}},
  title = {Proposition 16.3.2},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/C1971A5F3DFD/}
}

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Proposition 16.3.2C1971A5F3DFD

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Proposition 16.3.2`C1971A5F3DFD`