Proposition 11.1.16.2A8C104899E8 Suppose that every pair $u,v\in \mathcal H_\omega $ satisfies
F5F2F7988675
11.1.17
\begin{equation} \int _{\{ u<v\} }\omega _ v^ n \leq \int _{\{ u<v\} }\omega _ u^ n. \end{equation}
Then $\mathrm M_\omega $ is constant on $\mathcal H_\omega $.
Proof.
Fix $u,v\in \mathcal H_\omega $. By compactness there is a constant $C$ such that $u<v+C$ everywhere. Since $\omega _{v+C}=\omega _ v$, applying (11.1.17) to $u$ and $v+C$ gives $\mathrm M_\omega (v)\leq \mathrm M_\omega (u)$. Exchanging the roles of $u$ and $v$ gives the opposite inequality.
$\square$
Pome
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