Proposition 17.3.1292E951B4916A

Proof. If $u\leq u'$, every majorant of $u'$ in the definition is a majorant of $u$, so $E_\theta ^\phi (u)\leq E_\theta ^\phi (u')$. Translating majorants by a constant and using the shift formula on $\mathcal E^\infty (X,\theta ,\phi )$ gives the shift formula. The potentials $u_ k=\max \{ u,\phi -k\} $ belong to $\mathcal E^\infty (X,\theta ,\phi )$, decrease with $k$ and dominate $u$; hence $E_\theta ^\phi (u_ k)$ decreases and is bounded below by $E_\theta ^\phi (u)$. Conversely, let $w$ be any majorant in the definition, and choose $C$ with $w\geq \phi -C$. For $k\geq C$ both $u$ and $\phi -k$ lie below $w$, so $u_ k\leq w$ and $E_\theta ^\phi (u_ k)\leq E_\theta ^\phi (w)$. Letting $k\to \infty $ and taking the infimum over $w$ gives $\lim _ kE_\theta ^\phi (u_ k)\leq E_\theta ^\phi (u)$. $\square$