PomeProposition 16.23.2.E33C504A684F Let $\phi ,\psi \in \mathrm{PSH}(X,\theta )$ with $P_\theta [\phi ]=\phi $ and $P_\theta [\psi ]=\psi $, and suppose that $w=P_\theta (\phi ,\psi )\ne -\infty $. Then $P_\theta [w]=w$. Thus the rooftop envelope of two model potentials is a model potential whenever its mass is positive.
Proposition 16.23.2E33C504A684F
Proof. Since $w\leq \phi $, monotonicity gives $P_\theta [w]\leq P_\theta [\phi ]=\phi $, and likewise $P_\theta [w]\leq \psi $. Thus $P_\theta [w]$ is a common minorant of $\phi $ and $\psi $, and $P_\theta [w]\leq w$. Conversely, $w\leq \phi \leq 0$, so $w$ is a competitor for $P_\theta [w]$. $\square$
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