Lemma 17.4.10.F62E3A64C1EE For $u,v\in \mathcal E^1(X,\theta ,\phi )$,
\[ \int _ X(v-u)\, \mu _ v\leq E_\theta ^\phi (v)-E_\theta ^\phi (u) \leq \int _ X(v-u)\, \mu _ u. \]
If moreover $u\leq v$, then
\[ \frac1{n+1}\int _ X(v-u)\, \mu _ u\leq E_\theta ^\phi (v)-E_\theta ^\phi (u). \]
Proof.
The first display is the tangent bound (17.3.10), which holds on $\mathcal E^1(X,\theta ,\phi )$ by proposition 17.3.19. By the same proposition, the cocycle formula (17.3.6) holds for $u$ and $v$. If $u\leq v$, all its terms are nonnegative, and retaining the term $j=0$, whose measure is $\mu _ u$, gives the second display.
$\square$
Pome
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