Corollary 10.31.281B6B55C4EF1E

Proof. By theorem 10.31.22, there is $z\in \mathcal E_ V$ with $\mu _ z=\mathrm{e}^ z\nu $. Since $V$ is an ordinary model and $V\geq \varphi $, the preceding proposition applies with $\eta =V$ and gives $Q=P_\theta [\varphi ](z)\in \mathcal E_\varphi $. On the contact set $\{ Q=z\} $,

The weights are finite off the pole sets, which carry no mass. Thus $\mathrm{e}^{-Q}\mu _ Q\leq \nu $, and this measure is nonzero because $\mu _ Q(X)\geq \ell >0$. $\square$