Proposition 16.11.6046401771DAB

Proof. Commutativity, monotonicity and idempotence follow directly from the description of $P_\theta (u,v)$ as the largest common minorant. For associativity, let also $w\in \mathrm{PSH}(X,\theta )\cup \{ -\infty \} $. A potential lies below $P_\theta (u,v)$ and $w$ if and only if it lies below $u$, $v$ and $w$: when $P_\theta (u,v)\ne -\infty $ this is the maximality of $P_\theta (u,v)$, and otherwise neither condition can be satisfied. Hence $P_\theta (P_\theta (u,v),w)$ and $P_\theta (u,P_\theta (v,w))$ are both the largest common minorant of $u$, $v$ and $w$, or both equal $-\infty $ if there is none. The behavior under constant shifts and decreasing limits follows from proposition 16.11.4, applied to the obstacles $\min \{ u_ j,v_ j\} $, which are bounded above by $\sup _ Xu_1$ and decrease to $\min \{ u,v\} $. If $u$ or $v$ is identically $-\infty $, this limit obstacle has no competitor, in accordance with our convention.

For the last assertion put $B=\max \{ \sup _ Xu,-C\} $. Then $u-B\leq 0$, so $u-B\leq V_\theta $, and also $u-B\leq u+C$. Hence $u-B$ is a common minorant of $u+C$ and $V_\theta $, and