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Proposition 16.6.27F7C47263CB3C

Proposition 16.6.27.F7C47263CB3C

  1. 6B3F6E8F1EC1

    Suppose that $\varphi \in \mathrm{PSH}(X,\theta )$ and $\lambda \in \mathbb {R}_{>0}$. Then

    \[ P_{\lambda \theta }[\lambda \varphi ]_{\mathcal{I}}=\lambda P_{\theta }[\varphi ]_{\mathcal{I}}. \]
  2. 0F43574FD82C

    Suppose that $\varphi _1\in \mathrm{PSH}(X,\theta _1)$ and $\varphi _2\in \mathrm{PSH}(X,\theta _2)$. Then

    \[ P_{\theta _1+\theta _2}[\varphi _1+\varphi _2]_{\mathcal{I}}\geq P_{\theta _1}[\varphi _1]_{\mathcal{I}}+P_{\theta _2}[\varphi _2]_{\mathcal{I}}. \]
  3. 56C6F951EFD4

    Suppose that $\varphi _1\in \mathrm{PSH}(X,\theta _1)$ and $\varphi _2\in \mathrm{PSH}(X,\theta _2)$. Then

    \[ P_{\theta _1+\theta _2}[\varphi _1+\varphi _2]_{\mathcal{I}}\sim _{\mathcal{I}} P_{\theta _1}[\varphi _1]_{\mathcal{I}}+P_{\theta _2}[\varphi _2]_{\mathcal{I}}. \]
  4. 14196330587C

    Suppose that $\varphi _1,\varphi _2\in \mathrm{PSH}(X,\theta )$. Then

    \[ P_{\theta }[\varphi _1\lor \varphi _2]_{\mathcal{I}}\sim _{\mathcal{I}} P_{\theta }[\varphi _1]_{\mathcal{I}}\lor P_{\theta }[\varphi _2]_{\mathcal{I}}. \]

Proof. 1 This follows by rescaling the candidates in the definition of the $\mathcal{I}$-envelope.

2 Suppose that $\psi _1\in \mathrm{PSH}(X,\theta _1)$ and $\psi _2\in \mathrm{PSH}(X,\theta _2)$ satisfy

\[ \psi _ i\leq 0,\quad \psi _ i \sim _{\mathcal{I}} \varphi _ i \]

for $i=1,2$. Then thanks to proposition 16.6.5,

\[ \psi _1+\psi _2\leq 0,\quad \psi _1+\psi _2 \sim _{\mathcal{I}} \varphi _1+\varphi _2. \]

It follows that

\[ \psi _1+\psi _2\leq P_{\theta _1+\theta _2}[\varphi _1+\varphi _2]_{\mathcal{I}}. \]

Since $\psi _1$ and $\psi _2$ are arbitrary, we conclude.

3 and 4: These follow easily from proposition 16.6.11 and proposition 16.6.5. $\square$

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