Corollary 17.4.8838D06267D86

Proof. By proposition 17.4.7, the rooftop $w=P_\theta (u,v)$ belongs to $\mathcal E(X,\theta ,\phi )$. Since $w\preceq v$ and $\int _ X\langle \theta _ w^ n\rangle =\int _ X\langle \theta _ v^ n\rangle >0$, proposition 16.22.14 gives $P_\theta [w](v)=v$. Every competitor for $P_\theta [w](v)$ is more singular than $w$, hence than $u$, so it is a competitor for $P_\theta [u](v)$; and $P_\theta [u](v)\leq v$ quasi-everywhere by proposition 16.25.2(1), hence everywhere. Therefore $v=P_\theta [w](v)\leq P_\theta [u](v)\leq v$. $\square$