Proposition 18.28.6C9561913A3E8

Proposition 18.28.6.C9561913A3E8 Let $f,g\colon K\rightarrow [-\infty ,\infty ]$ be functions and $\varphi ,\psi \in \mathrm{PSH}(X,\theta )$. Then we have the following:

1. 33B49F7D21A7

   We have

   \[ P_{\theta ,K}[\varphi ](f)\leq P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f); \]
2. 6B1193ABBD59

   assume that $f$ is bounded from above, then

   \[ \begin{aligned} \left.P_{\theta ,K}[\varphi ](f)\right|_ K\leq & f\textup{ quasi-everywhere},\\ \left. P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\right|_ K\leq & f\textup{ quasi-everywhere}; \end{aligned} \]
3. 27E6FD2D7332

   suppose that $\varphi \preceq _{\mathcal{I}}\psi $, then

   \[ P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\leq P_{\theta ,K}[\psi ]_{\mathcal{I}}(f); \]
4. B3B86BE1530D

   if $f\leq g$, then

   \[ P_{\theta ,K}[\varphi ](f)\leq P_{\theta ,K}[\varphi ](g),\quad P_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)\leq P_{\theta ,K}[\varphi ]_{\mathcal{I}}(g); \]
5. 13DB8823929A

   for any $t \in (0,1)$, we have

   \[ \begin{aligned} P_{\theta ,K}\left[t\varphi +(1-t)\psi \right](f)\geq & tP_{\theta ,K}[\varphi ](f)+(1-t)P_{\theta ,K}[\psi ](f),\\ P_{\theta ,K}\left[t\varphi +(1-t)\psi \right]_{\mathcal{I}}(f)\geq & tP_{\theta ,K}[\varphi ]_{\mathcal{I}}(f)+(1-t)P_{\theta ,K}[\psi ]_{\mathcal{I}}(f). \end{aligned} \]