Corollary 17.15.17.C8D1BA787705 Let $\Gamma ,\Gamma '\in \mathrm{TC}(X,\theta )_{>0}$. Then $\Gamma \lor \Gamma '\in \mathrm{TC}(X,\theta )_{>0}$ and
9AB9F5A23D95
17.15.18
\begin{equation} (\Gamma \lor \Gamma ')_{-\infty }=P_{\theta }\left[\Gamma _{-\infty }\lor \Gamma '_{-\infty }\right]. \end{equation}
If $\Gamma ,\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$, then $\Gamma \lor \Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$.
For each $\Gamma ''\in \mathrm{TC}(X,\theta )_{>0}$ and each $\Gamma ''\geq \Gamma $ and $\Gamma ''\geq \Gamma '$, we have $\Gamma ''\geq \Gamma \lor \Gamma '$.
Moreover, the operation $\lor $ is associative and commutative.
Proof.
It follows immediately from 17.15.14 that $\Gamma \lor \Gamma '\in \mathrm{TC}(X,\theta )_{>0}$, and it lies in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ if $\Gamma ,\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$.
We argue (17.15.18). Set
\[ \eta \coloneqq \Gamma _{-\infty }\lor \Gamma '_{-\infty }. \]
Since $\Gamma \lor \Gamma '\geq \Gamma $ and $\Gamma \lor \Gamma '\geq \Gamma '$, we have $(\Gamma \lor \Gamma ')_{-\infty }\geq \eta $. As $(\Gamma \lor \Gamma ')_{-\infty }$ is a model potential, it follows that
\[ (\Gamma \lor \Gamma ')_{-\infty }\geq P_\theta [\eta ]. \]
Conversely, for each $\rho \in \mathbb {R}$ we have $\Gamma _\rho \lor \Gamma '_\rho \leq \eta $. Hence the constant curve with value $\eta $ is a concave majorant of $\rho \mapsto \Gamma _\rho \lor \Gamma '_\rho $, and therefore
\[ \operatorname{CE}\left(\rho \mapsto \Gamma _\rho \lor \Gamma '_\rho \right)_\tau \leq \eta ,\quad \tau <\Gamma _{\max }\lor \Gamma '_{\max }. \]
Applying the monotonicity of the $P$-envelope and then letting $\tau \to -\infty $ gives
\[ (\Gamma \lor \Gamma ')_{-\infty }\leq P_\theta [\eta ]. \]
This proves (17.15.18).
Take $\Gamma ''$ as in the statement of the corollary. First observe that
\[ \Gamma ''_{\max }\geq \Gamma _{\max } \lor \Gamma '_{\max }=(\Gamma \lor \Gamma ')_{\max }. \]
Take $\tau <(\Gamma \lor \Gamma ')_{\max }$. We argue that
\[ \Gamma ''_{\tau }\geq (\Gamma \lor \Gamma ')_{\tau }. \]
By the concavity of $\Gamma ''$, the potential $\Gamma ''_\tau $ dominates the concave envelope
\[ \operatorname{CE}\left(\rho \mapsto \Gamma _{\rho }\lor \Gamma '_{\rho }\right)_\tau . \]
Since $\Gamma ''_\tau $ is model, the monotonicity of the $P$-envelope gives
\[ \Gamma ''_{\tau }\geq (\Gamma \lor \Gamma ')_{\tau }. \]
Therefore,
\[ \Gamma ''\geq \Gamma \lor \Gamma '. \]
Commutativity is immediate from the definition. Associativity follows from the least-upper-bound property just proved: both $(\Gamma \lor \Gamma ')\lor \Gamma ''$ and $\Gamma \lor (\Gamma '\lor \Gamma '')$ are the least upper bound of $\Gamma ,\Gamma ',\Gamma ''$.
$\square$
Pome
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