Proposition 16.3.9EF5AC4B19C4D

Proof. By symmetry, it suffices to show that

We may assume that $g\geq 0$ (after replacing $g$ by $g+C$ with $C\geq -\inf _ X g$, which leaves $\theta '$ unchanged and shifts $\varphi '$ by an additive constant). Then for any $\psi \in \mathrm{PSH}(X,\theta )$ with $\psi \preceq \varphi $ and $\psi \leq 0$, we set $\psi '\coloneqq \psi -g\in \mathrm{PSH}(X,\theta ')$. Then $\psi '\preceq \varphi '$ and $\psi '\leq 0$, so $\psi '\leq P_{\theta '}[\varphi ']$. Since $\psi $ is arbitrary, it follows that

which gives $P_{\theta }[\varphi ]\preceq P_{\theta '}[\varphi ']$. $\square$