PomeDefinition 16.25.1.3558A6E66BA8 For $u\in \mathrm{PSH}(X,\theta )$ and a Borel obstacle $f$, define
\[ P_\theta [u](f)=\left(\sup \{ v\in \mathrm{PSH}(X,\theta ): v\preceq u,\ v\leq f\text{ quasi-everywhere}\} \right)^* \]
when this family is nonempty and uniformly bounded above, and put $P_\theta [u](f)=-\infty $ when it is empty. For Borel obstacles $g$ and $f$ we write $P_\theta (g,f)=P_\theta (\min \{ g,f\} )$, under the same conditions on the family of competitors.
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