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Proposition 16.5.3651CFFEE97C34

Proposition 16.5.36.51CFFEE97C34 Let $\varphi ,\psi \in \mathcal{E}^1(X,\theta ;\phi )$. Then

3.132
\begin{equation} \int _ X \left(\psi -\varphi \right)\theta ^ n_{\psi }\leq E_{\theta }^{\phi }(\psi )-E_{\theta }^{\phi }(\varphi )\leq \int _ X \left(\psi -\varphi \right)\theta ^ n_{\varphi }. \end{equation}

Proof. Thanks to (16.5.32), this can be reduced to the case where $\varphi ,\psi \in \mathcal{E}^{\infty }(X,\theta ;\phi )$, which is established in 16.5.16. $\square$

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