Proposition 17.4.62FA1EC20E3A2

Proof. By lemma 16.22.2, every member of the family is bounded above by $\phi +C$ for a common constant $C$, and so is the regularized upper envelope $U$. Fix a member $u_0$ of the family; then $u_0\leq U\preceq \phi $. For $\mathcal F=\mathcal E(X,\theta ,\phi )$, mass monotonicity gives $m=\int _ X\langle \theta _{u_0}^ n\rangle \leq \int _ X\langle \theta _ U^ n\rangle \leq m$. For $\mathcal F=\mathcal E^1(X,\theta ,\phi )$, the last assertion of corollary 17.3.17, applied to $u_0\leq U$, gives $U\in \mathcal F$. For $\mathcal F=\mathcal E^\infty (X,\theta ,\phi )$, the member $u_0$ supplies a lower bound $U\geq u_0\geq \phi -C'$. $\square$