Proposition 16.23.337241FAAC9EF

Proof. Shifting $u$ and $v$ by constants affects neither the existence of a common minorant nor the mass of $P_\theta (u,v)$. Indeed, a common minorant can be shifted down by the largest absolute change, and by proposition 16.11.4(4) the rooftop envelopes of the original and of the shifted pair differ by a bounded amount, because their obstacles do. We may therefore assume that $u\leq \phi $ and $v\leq \psi $.

We first replace $\phi $ by $u$, and then $\psi $ by $v$; both steps combine extrapolation with proposition 16.23.1. Since $w\leq \phi $ and $w\leq \psi $, the masses of $\phi $ and $\psi $ are at least that of $w$, hence positive. For $a>1$ let $u_ a=E_ a(u,\phi )$. Since $u$ and $\phi $ have the same positive mass, theorem 16.22.4 shows that $u_ a$ also has this mass and that $a^{-1}u_ a+(1-a^{-1})\phi \leq u$; moreover $u_ a\leq u\leq \phi $, because the obstacle $u+(a-1)(u-\phi )$ of $u_ a$ is at most $u$. Since $\max \{ u_ a,w\} \leq \phi $ and $\int _ X\langle \theta _ w^ n\rangle >0$,

By proposition 16.23.1, the rooftop envelope $P_\theta (u_ a,w)$ is a potential. By associativity and $u_ a\leq \phi $, it equals $P_\theta (u_ a,\psi )$. The potential

lies below $a^{-1}u_ a+(1-a^{-1})\phi \leq u$ and below $\psi $, so it is a common minorant of $u$ and $\psi $. Hence $z=P_\theta (u,\psi )$ is a potential, and positivity and monotonicity of mass give $\int _ X\langle \theta _ z^ n\rangle \geq (1-a^{-1})^ n\int _ X\langle \theta _ w^ n\rangle $. On the other hand $z\leq P_\theta (\phi ,\psi )=w$, so the mass of $z$ is at most that of $w$. Letting $a\to \infty $, we see that $z$ and $w$ have the same positive mass.

Now let $v_ a=E_ a(v,\psi )$. In the same way, $v_ a\leq v\leq \psi $, the mass of $v_ a$ is that of $\psi $, and $a^{-1}v_ a+(1-a^{-1})\psi \leq v$. Since $\max \{ v_ a,z\} \leq \psi $ and $z$ has positive mass, we have

so proposition 16.23.1 shows that $P_\theta (z,v_ a)$ is a potential, and by associativity and $v_ a\leq \psi $ it equals $P_\theta (u,v_ a)$. The potential

lies below $u$ and below $a^{-1}v_ a+(1-a^{-1})\psi \leq v$, and its mass is at least $(1-a^{-1})^ n\int _ X\langle \theta _ z^ n\rangle $. Hence $P_\theta (u,v)$ is a potential whose mass is at least $(1-a^{-1})^ n\int _ X\langle \theta _ w^ n\rangle $, for every $a>1$. Since $P_\theta (u,v)\leq w$, its mass is at most that of $w$, and letting $a\to \infty $ completes the proof. $\square$