Lemma 16.5.16.8D9EA7C5CC7C Let $\varphi ,\psi \in \mathcal{E}^{\infty }(X,\theta ;\phi )$. Then
6E4685F8B030
16.5.17
\begin{equation} \int _ X (\varphi -\psi )\, \theta _{\psi }^ n \geq E_{\theta }^{\phi }(\varphi )-E_{\theta }^{\phi }(\psi )\geq \int _ X (\varphi -\psi )\, \theta _{\varphi }^ n. \end{equation}
Proof.
Thanks to (16.5.13), we may assume that $\varphi \geq \psi $.
As we have seen in (16.5.15), the middle term can be written as
\[ \frac{1}{n+1}\sum _{j=0}^ n \int _ X(\varphi -\psi )\, \theta _{\varphi }^ j\wedge \theta _{\psi }^{n-j}. \]
We claim that for each individual $j$, the inequality (16.5.17) holds:
\[ \int _ X (\varphi -\psi )\, \theta _{\psi }^ n \geq \int _ X(\varphi -\psi )\, \theta _{\varphi }^ j\wedge \theta _{\psi }^{n-j} \geq \int _ X (\varphi -\psi )\, \theta _{\varphi }^ n. \]
Thanks to proposition 16.5.5, we have
\[ \begin{aligned} \int _ X(\varphi -\psi )\, \theta _{\varphi }^ j\wedge \theta _{\psi }^{n-j}=& \int _0^{\infty } \int _{\{ \varphi >\psi +t\} } \theta _{\varphi }^ j\wedge \theta _{\psi }^{n-j}\, \mathrm{d}t\\ \geq & \int _0^{\infty } \int _{\{ \varphi >\psi +t\} } \theta _{\varphi }^ n\, \mathrm{d}t \\ =& \int _ X (\varphi -\psi )\, \theta _{\varphi }^ n. \end{aligned} \]
The other inequality is similar.
$\square$
Pome
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