PomeProposition 17.10.39.FD1B5378AA06 Let $\ell \in \mathcal{R}(X,\theta ;\phi )$. Then there is a constant $C\in \mathbb {R}$ such that
\[ \sup _{X}\ell _ t=Ct,\quad t\geq 0. \]
Proposition 17.10.39FD1B5378AA06
Proof. It follows from proposition 17.10.29 that
\[ \sup _{\{ \phi \neq -\infty \} }(\ell _ t-\phi )=t\sup _{\{ \phi \neq -\infty \} }(\ell _1-\phi ) \]
for all $t\geq 0$.
It suffices to show that for any $t\geq 0$,
\[ \sup _{\{ \phi \neq -\infty \} }(\ell _ t-\phi )=\sup _ X \ell _ t. \]
This was already proved in proposition 16.3.21. $\square$
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