Proposition 17.15.7.FBC25DBE6122 Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$ and $\Gamma ' \in \mathrm{TC}(X,\theta ')_{>0}$. Then $\Gamma +\Gamma '\in \mathrm{TC}(X,\theta +\theta ')_{>0}$. Moreover,
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17.15.8
\begin{equation} (\Gamma +\Gamma ')_{-\infty }=P_{\theta +\theta '}\left[\Gamma _{-\infty }+\Gamma '_{-\infty }\right]. \end{equation}
When $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ and $\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta ')_{>0}$, we have $\Gamma +\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta +\theta ')_{>0}$.
The operation $+$ is commutative and associative.
Proof.
It follows immediately from 17.15.6 that $\Gamma +\Gamma '\in \mathrm{TC}(X,\theta +\theta ')_{>0}$. It lies in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta +\theta ')_{>0}$ if $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ and $\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta ')_{>0}$.
We argue (17.15.8). By definition, for any small enough $\tau $, we have
\[ (\Gamma +\Gamma ')_{-\infty }\geq (\Gamma +\Gamma ')_{2\tau }\succeq _ P \Gamma _{\tau }+\Gamma '_{\tau }. \]
Letting $\tau \to -\infty $ and applying proposition 16.9.44 and theorem 16.9.45, we find that
\[ (\Gamma +\Gamma ')_{-\infty }\succeq _ P \Gamma _{-\infty }+\Gamma '_{-\infty }. \]
On the other hand, for each small enough $\tau $, we have
\[ (\Gamma +\Gamma ')_{\tau }\sim _ P \sup _{\delta \in \mathbb {R}}\left( \Gamma _\delta +\Gamma '_{\tau -\delta }\right) \preceq _ P \Gamma _{-\infty }+\Gamma '_{-\infty } \]
by proposition 16.8.27 and proposition 16.9.44. We apply proposition 16.9.44 again, we conclude that
\[ (\Gamma +\Gamma ')_{-\infty } \preceq _ P \Gamma _{-\infty }+\Gamma '_{-\infty }. \]
So (17.15.8) follows.
Finally, let us show that $+$ is commutative and associative. Commutativity follows from the symmetry of the defining supremum. Let $\Gamma ''\in \mathrm{TC}(X,\theta '')_{>0}$. Then we want to show that
\[ (\Gamma +\Gamma ')+\Gamma ''=\Gamma +(\Gamma '+\Gamma ''). \]
First observe that
\[ \left((\Gamma +\Gamma ')+\Gamma '' \right)_{\max }=\left( \Gamma +(\Gamma '+\Gamma '')\right)_{\max }. \]
Fix $\tau $ less than this common value. We compute that
\[ \begin{aligned} & \left( (\Gamma +\Gamma ')+\Gamma ''\right)_{\tau }\\ =& P_{\theta +\theta '+\theta ''}\left[\sup _{\delta _1\in \mathbb {R}} \left((\Gamma +\Gamma ')_{\delta _1}+\Gamma ''_{\tau -\delta _1}\right)\right]\\ \sim _ P & \sup _{\delta _1\in \mathbb {R}} \left((\Gamma +\Gamma ')_{\delta _1}+\Gamma ''_{\tau -\delta _1}\right)\\ \sim _ P & \sup _{\delta _1,\delta _2\in \mathbb {R}} \left(\Gamma _{\delta _2}+\Gamma '_{\delta _1-\delta _2}+\Gamma ''_{\tau -\delta _1}\right), \end{aligned} \]
where in the last line, we applied proposition 16.9.44 and proposition 16.8.27. Similarly, for $\left( \Gamma +(\Gamma '+\Gamma '')\right)_{\tau }$, we get the same expression. The associativity follows.
$\square$
Pome
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