Proposition 10.31.254F5454D09F78

Proof. First suppose $z\leq \eta $; recall also that $\varphi \leq \eta \leq 0$. For $b>1$ sufficiently close to one, lemma 13.20.1 gives

which is not identically $-\infty $. Set $a=1-b^{-1}$. The stretching lemma lemma 16.28.63, now applied in $\mathcal E_\eta $, gives $z_{1/a}=Q_{1/a}(z,\eta )\in \mathcal E_\eta $. The potential

lies below both $\varphi $ and $z$. Indeed, $z_{1/a}\leq \eta $ and $(1-a)h_ b+a\eta \leq \varphi $ give the first bound, while $h_ b\leq \eta $ and $a z_{1/a}+(1-a)\eta \leq z$ give the second. Thus $p=P_\theta (\varphi ,z)$ is not identically $-\infty $. Volume addition and monotonicity give

As $b\downarrow 1$, we have $a\downarrow 0$, and the first term tends to the last by lemma 13.20.1. The upper volumes of $p$ and $\varphi $ are therefore equal. Since $p\leq \varphi $, theorem 13.18.5 gives $p\in \mathcal E_\varphi $.

For general $z$, the envelope identity in $\mathcal E_\eta $ gives $z-\sup _ Xz\leq \eta $. Set $S=\max (\sup _ Xz,0)$. The preceding case gives $P_\theta (\varphi ,z-S)\in \mathcal E_\varphi $, and

Upward stability finishes the proof. $\square$