Lemma 16.5.43.BAAC43C43BD5 Let $\varphi ,\psi ,\gamma \in \mathcal{E}(X,\theta ;\phi )$. Assume that $\varphi \leq \psi \leq \gamma $. Then
\[ \int _ X (\gamma -\psi )\, \theta _{\psi }^ n\leq 2^{n+1}\int _ X (\gamma -\varphi )\, \theta _{\varphi }^ n. \]
Proof.
Observe that for any $t\geq 0$,
7058C51CA443
16.5.44
\begin{equation} \begin{split} \{ \gamma >\psi +2t\} \setminus \{ \psi =-\infty \} \subseteq \left\{ (\gamma +\psi )/2>\psi +t\right\} \\ \subseteq \left\{ (\gamma +\psi )/2>\varphi +t\right\} \subseteq \{ \gamma >\varphi +t\} . \end{split} \end{equation}
So
\[ \begin{aligned} \int _ X (\gamma -\psi )\, \theta _{\psi }^ n =& 2\int _0^{\infty } \int _{\{ \gamma -\psi >2t\} }\, \theta _{\psi }^ n\, \mathrm{d}t & \\ \leq & 2\int _0^{\infty } \int _{\left\{ (\gamma +\psi )/2>\psi +t\right\} }\, \theta _{\psi }^ n\, \mathrm{d}t & \textup{by (\href{../../tag/7058C51CA443/}{16.5.44})}\\ \leq & 2^{n+1}\int _0^{\infty } \int _{\left\{ (\gamma +\psi )/2>\varphi +t\right\} }\, \theta _{(\gamma +\psi )/2}^ n\, \mathrm{d}t & \\ \leq & 2^{n+1}\int _0^{\infty } \int _{\left\{ (\gamma +\psi )/2>\varphi +t\right\} }\, \theta _{\varphi }^ n\, \mathrm{d}t & \textup{by proposition \href{../../tag/FDFD0A6F6287/}{16.5.5}}\\ \leq & 2^{n+1}\int _0^{\infty } \int _{\left\{ \gamma >\varphi +t\right\} }\, \theta _{\varphi }^ n\, \mathrm{d}t & \textup{by (\href{../../tag/7058C51CA443/}{16.5.44})}\\ =& 2^{n+1} \int _ X (\gamma -\varphi )\, \theta _{\varphi }^ n & . \end{aligned} \]
This completes the proof.
$\square$
Pome
Comments
Corrections, questions, and mathematical discussion are welcome. Sign in with GitHub to comment. Comments are public.