PomeDefinition 17.15.1.DF47E4B2060A Given $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$, $\Gamma '\in \mathrm{TC}(X,\theta ')_{>0}$, we say $\Gamma \leq \Gamma '$ if $\Gamma _{\max }\leq \Gamma '_{\max }$ and, for all $\tau <\Gamma _{\max }$, we have
17.15 Operations on test curves9D751AA26579
9D751AA26579
Let $X$ be a connected compact Kähler manifold of dimension $n$ and $\theta ,\theta ',\theta ''$ be smooth closed real $(1,1)$-forms on $X$ representing big cohomology classes.
In this section, we develop several general operations on test curves, anticipating the applications in non-Archimedean geometry in section 22.1. One may read section 22.1 first and consult this section when necessary.
Lemma 17.15.3.CA55D62C74E0 Let $\Gamma ,\Gamma '\in \mathrm{TC}(X,\theta )_{>0}$ and $\omega $ be a closed real smooth positive $(1,1)$-form on $X$. Then the following are equivalent:
7981AA897F00$\Gamma \leq \Gamma '$;
ED51882DAEE9$P_{\theta +\omega }[\Gamma ]\leq P_{\theta +\omega }[\Gamma ']$.
Proof. This follows from example 16.8.6. $\square$
Definition 17.15.4.DAA4083947C1 Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$ and $\Gamma ' \in \mathrm{TC}(X,\theta ')_{>0}$. Then we define $\Gamma +\Gamma '\in \mathrm{TC}(X,\theta +\theta ')_{>0}$ as follows:
D6CB3C2695C3We set
\[ (\Gamma +\Gamma ')_{\max }\coloneqq \Gamma _{\max }+\Gamma '_{\max }; \]B7E27AEC255Cfor any $\tau <(\Gamma +\Gamma ')_{\max }$, we define2
6C3AF1760F9F17.15.5\begin{equation} (\Gamma +\Gamma ')_{\tau }\coloneqq P_{\theta +\theta '}\left[\sup _{\delta \in \mathbb {R}} \left(\Gamma _\delta +\Gamma '_{\tau -\delta }\right)\right]. \end{equation}
Lemma 17.15.6.5BE2CC04AAD0 Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$ and $\Gamma ' \in \mathrm{TC}(X,\theta ')_{>0}$. Then we have $\Gamma +\Gamma '\in \mathrm{TC}(X,\theta +\theta ')_{>0}$. In particular, for any $\tau <(\Gamma +\Gamma ')_{\max }$,
\[ \sup _{\delta \in \mathbb {R}} \left(\Gamma _\delta +\Gamma '_{\tau -\delta }\right)\in \mathrm{PSH}(X,\theta +\theta '). \]
This potential is $\mathcal{I}$-good if $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ and $\Gamma ' \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta ')_{>0}$.
Proof. Let
\[ \eta _{\tau }=\sup _{\delta \in \mathbb {R}} \left(\Gamma _\delta +\Gamma '_{\tau -\delta }\right)=\sup _{\tau -\Gamma '_{\max }<\delta <\Gamma _{\max }}\left(\Gamma _\delta +\Gamma '_{\tau -\delta }\right) \]
for all $\tau \in \mathbb {R}$. Set
\[ Z=\left\{ x\in X: \Gamma _{\upsilon }(x)=-\infty \, \forall \upsilon \in \mathbb {R}\right\} \cup \left\{ x\in X: \Gamma '_{\upsilon }(x)=-\infty \, \forall \upsilon \in \mathbb {R}\right\} . \]
It follows from proposition 21.11.37 that for any $x\in X\setminus Z$, we have
\[ \eta ^*_ t(x)=\Gamma ^*_ t(x)+\Gamma '^*_ t(x) \]
for all $t>0$. The same trivially holds when $x\in Z$, so the equation holds everywhere. In particular, by corollary 21.11.26 and proposition 5.14.15, we have
\[ \eta _{\tau }=\left(\Gamma ^*+\Gamma '^*\right)^*_{\tau }\in \mathrm{PSH}(X,\theta +\theta ') \]
when $\tau <\Gamma _{\max }+\Gamma '_{\max }$.
We now check that $\Gamma +\Gamma '$ is a test curve. The family $\tau \mapsto \eta _\tau $ is decreasing, and the monotonicity of the $P$-envelope proposition 16.4.2 3 gives the decreasingness of $\Gamma +\Gamma '$. If $\tau =(1-s)\tau _0+s\tau _1$, then for arbitrary $\delta _0,\delta _1\in \mathbb {R}$ and $\delta =(1-s)\delta _0+s\delta _1$,
\[ \Gamma _\delta +\Gamma '_{\tau -\delta }\geq (1-s)(\Gamma _{\delta _0}+\Gamma '_{\tau _0-\delta _0})+s(\Gamma _{\delta _1}+\Gamma '_{\tau _1-\delta _1}). \]
Taking suprema and using proposition 16.4.2 gives the concavity of $\Gamma +\Gamma '$.
Fix $\tau <\Gamma _{\max }+\Gamma '_{\max }$. Choose $\delta \in \mathbb {R}$ with $\delta <\Gamma _{\max }$ and $\tau -\delta <\Gamma '_{\max }$. Then
\[ \eta _\tau \geq \Gamma _\delta +\Gamma '_{\tau -\delta }. \]
By 17.12.6, $\Gamma _\delta \in \mathrm{PSH}(X,\theta )_{>0}$ and $\Gamma '_{\tau -\delta }\in \mathrm{PSH}(X,\theta ')_{>0}$. Hence $\Gamma _\delta +\Gamma '_{\tau -\delta }$ has positive $(\theta +\theta ')$-mass by the multi-linearity of non-pluripolar products proposition 13.11.4 6. It follows from the monotonicity theorem theorem 13.13.11 that $\eta _\tau \in \mathrm{PSH}(X,\theta +\theta ')_{>0}$. By corollary 16.3.20,
\[ (\Gamma +\Gamma ')_\tau =P_{\theta +\theta '}[\eta _\tau ] \]
is a model potential. Finally, $(\Gamma +\Gamma ')_{-\infty }\in \mathrm{PSH}(X,\theta +\theta ')$ by proposition 5.14.2; since it dominates $(\Gamma +\Gamma ')_\tau $ for any $\tau <\Gamma _{\max }+\Gamma '_{\max }$, it has positive mass by another application of the monotonicity theorem theorem 13.13.11. Thus $\Gamma +\Gamma '\in \mathrm{TC}(X,\theta +\theta ')_{>0}$.
Next, assume that $\Gamma $ and $\Gamma '$ are $\mathcal{I}$-model. Fix $\tau <\Gamma _{\max }+\Gamma '_{\max }$. Then for each $\delta \in \mathbb {R}$ such that $\delta <\Gamma _{\max }$ and $\tau -\delta <\Gamma '_{\max }$, we know that $\Gamma _\delta \in \mathrm{PSH}(X,\theta )_{>0}$ and $\Gamma '_{\tau -\delta }\in \mathrm{PSH}(X,\theta ')_{>0}$ by 17.12.6. It follows from example 18.1.11 that $\Gamma _\delta $ and $\Gamma '_{\tau -\delta }$ are both $\mathcal{I}$-good, hence so is $\Gamma _\delta +\Gamma '_{\tau -\delta }\in \mathrm{PSH}(X,\theta +\theta ')_{>0}$ by proposition 18.1.17. These sums are bounded above by $0$, so $\eta _{\tau }$ is $\mathcal{I}$-good by proposition 18.1.19. Since $(\Gamma +\Gamma ')_\tau =P_{\theta +\theta '}[\eta _\tau ]$, we conclude that $\Gamma +\Gamma '$ is $\mathcal{I}$-model. $\square$
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,
22BECA7DAF92
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. $\square$
Lemma 17.15.9.8571F1340B16 Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$ and $\Gamma ' \in \mathrm{TC}(X,\theta ')_{>0}$. Then for any closed smooth positive $(1,1)$-forms $\omega $ and $\omega '$ on $X$, we have
\[ P_{\theta +\theta '+\omega +\omega '}[\Gamma +\Gamma ']=P_{\theta +\omega }[\Gamma ]+P_{\theta '+\omega '}[\Gamma ']. \]
Proof. Observe that
\[ \begin{aligned} P_{\theta +\theta '+\omega +\omega '}[\Gamma +\Gamma ']_{\max }=& \left(P_{\theta +\omega }[\Gamma ]+P_{\theta '+\omega '}[\Gamma '] \right)_{\max }\\ =& \Gamma _{\max }+\Gamma '_{\max }. \end{aligned} \]
Take $\tau \in \mathbb {R}$ less than this common value. We need to verify that
\[ (\Gamma +\Gamma ')_{\tau }\sim _ P \Bigl ( P_{\theta +\omega }[\Gamma ]+P_{\theta '+\omega '}[\Gamma ']\Bigr )_{\tau }. \]
By definition, this means that
\[ \sup _{\tau -\Gamma '_{\max }<\delta <\Gamma _{\max }}\left(\Gamma _{\delta }+\Gamma '_{\tau -\delta }\right)\sim _ P \sup _{\tau -\Gamma '_{\max }<\delta <\Gamma _{\max }}\left(P_{\theta +\omega }[\Gamma _{\delta }]+P_{\theta '+\omega '}[\Gamma '_{\tau -\delta }]\right). \]
This is a consequence of proposition 16.8.27 and proposition 16.8.28. $\square$
Definition 17.15.10.7610A81B88AA Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$ and $C\in \mathbb {R}$, we define $\Gamma +C\in \mathrm{TC}(X,\theta )_{>0}$ as follows:
29950FACA7ABWe set
\[ (\Gamma +C)_{\max }\coloneqq \Gamma _{\max }+C; \]50DEF1EDA999for any $\tau <(\Gamma +C)_{\max }$, we set
\[ (\Gamma +C)_{\tau }\coloneqq \Gamma _{\tau -C}. \]
Proposition 17.15.11.6DDF81D7BEFF Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$, $\Gamma '\in \mathrm{TC}(X,\theta ')_{>0}$ and $C,C'\in \mathbb {R}$. Then
7D5EA90D003B$(\Gamma +\Gamma ')+C=\Gamma +(\Gamma '+C)=(\Gamma +C)+\Gamma '$;
01CBA28FD511$\Gamma +(C+C')=(\Gamma +C)+C'$.
Proof. 1 We first observe that
\[ \left((\Gamma +\Gamma ')+C\right)_{\max }=\left( \Gamma +(\Gamma '+C) \right)_{\max }=\left( (\Gamma +C)+\Gamma ' \right)_{\max }=\Gamma _{\max }+\Gamma '_{\max }+C. \]
Take any $\tau \in \mathbb {R}$ less than this common value. We compute
\[ \begin{aligned} \left((\Gamma +\Gamma ')+C\right)_{\tau }=& (\Gamma +\Gamma ')_{\tau -C} = P_{\theta +\theta '}\left[ \sup _{\delta \in \mathbb {R}}\left( \Gamma _\delta +\Gamma '_{\tau -C-\delta } \right) \right],\\ \left( \Gamma +(\Gamma '+C) \right)_{\tau }=& P_{\theta +\theta '}\left[ \sup _{\delta \in \mathbb {R}}\left( \Gamma _\delta +(\Gamma '+C)_{\tau -\delta } \right) \right]\\ =& P_{\theta +\theta '}\left[ \sup _{\delta \in \mathbb {R}}\left( \Gamma _\delta +\Gamma '_{\tau -C-\delta } \right) \right],\\ \left((\Gamma +C)+\Gamma '\right)_{\tau }=& P_{\theta +\theta '}\left[ \sup _{\delta \in \mathbb {R}}\left( (\Gamma +C)_{C+\delta }+\Gamma '_{\tau -C-\delta } \right) \right]\\ =& P_{\theta +\theta '}\left[ \sup _{\delta \in \mathbb {R}}\left( \Gamma _\delta +\Gamma '_{\tau -C-\delta } \right) \right]. \end{aligned} \]
2 Observe that
\[ \left( \Gamma +(C+C') \right)_{\max }=\left( (\Gamma +C)+C' \right)_{\max }=\Gamma _{\max }+C+C'. \]
For any $\tau \in \mathbb {R}$ less than this value, we have
\[ \left( \Gamma +(C+C') \right)_{\tau }=\Gamma _{\tau -C-C'}=\left( (\Gamma +C)+C' \right)_{\tau }. \]
This completes the proof. $\square$
Definition 17.15.12.D347D46F3537 Let $\Gamma ,\Gamma '\in \mathrm{TC}(X,\theta )_{>0}$. We define $\Gamma \lor \Gamma '\in \mathrm{TC}(X,\theta )_{>0}$ as follows:
12F2D0314E20We set
\[ (\Gamma \lor \Gamma ')_{\max }\coloneqq \Gamma _{\max }\lor \Gamma '_{\max }; \]08596243109Dfor any $\tau <(\Gamma \lor \Gamma ')_{\max }$, we define
CFA9EE6DC1ED17.15.13\begin{equation} (\Gamma \lor \Gamma ')_{\tau }\coloneqq P_{\theta }\left[ \operatorname{CE}\left( \rho \mapsto \Gamma _{\rho }\lor \Gamma '_{\rho } \right)_{\tau } \right]. \end{equation}
On the right-hand side of (17.15.13), we first take the upper concave envelope $\operatorname{CE}\left( \rho \mapsto \Gamma _{\rho }\lor \Gamma '_{\rho } \right)$, this gives a family of functions indexed by $\rho \in \mathbb {R}$. Then the notation $\operatorname{CE}\left( \rho \mapsto \Gamma _{\rho }\lor \Gamma '_{\rho } \right)_{\tau }$ denotes the function obtained by evaluating this family at $\tau $.
Lemma 17.15.14.77520F48ECEB Let $\Gamma ,\Gamma '\in \mathrm{TC}(X,\theta )_{>0}$. Then for any $\tau <\Gamma _{\max }\lor \Gamma '_{\max }$, we have
\[ \operatorname{CE}\left( \rho \mapsto \Gamma _{\rho }\lor \Gamma '_{\rho } \right)_{\tau }\in \mathrm{PSH}(X,\theta ), \]
and $\Gamma \lor \Gamma '\in \mathrm{TC}(X,\theta )_{>0}$.
This potential is $\mathcal{I}$-good if $\Gamma ,\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$.
In particular, (17.15.13) in definition 17.15.12 makes sense.
Proof. To simplify the notation, we write
\[ \psi _{\tau }=\operatorname{CE}\left( \rho \mapsto \Gamma _{\rho }\lor \Gamma '_{\rho } \right)_{\tau } \]
for all $\tau \in \mathbb {R}$. Thanks to proposition 21.11.36, for any fixed $x\in X$, we have
B75DA94B0995
17.15.15
\begin{equation} \psi ^*_ t(x)=\Gamma ^*_ t(x)\lor \Gamma '^*_ t(x) \end{equation}
for all $t>0$ as long as $\Gamma _{\tau }(x)\neq -\infty $ and $\Gamma '_{\tau }(x)\neq -\infty $ for some $\tau \in \mathbb {R}$. Otherwise, assume for instance that $\Gamma _{\tau }(x)=-\infty $ for all $\tau \in \mathbb {R}$; then by definition, $\psi _{\tau }(x)=\Gamma '_{\tau }(x)$ for all $\tau \in \mathbb {R}$. Therefore, $\Gamma ^*_ t(x)=-\infty $ for all $t>0$ and (17.15.15) continues to hold. The case where $\Gamma '_{\tau }(x)=-\infty $ for all $\tau $ is symmetric. Therefore, we have shown that
\[ \psi ^*_ t=\Gamma ^*_ t\lor \Gamma '^*_ t\in \mathrm{PSH}(X,\theta ). \]
It follows from proposition 17.10.13 that $(\psi ^*_ t)_{t\in [a,b]}$ is a subgeodesic for any $0< a<b$.
Next we observe that $\psi _{\bullet }$ is closed by definition. So it follows from proposition 21.11.36 and proposition 5.14.15 that
\[ \psi _{\tau }=(\psi ^*_{\bullet })^*_{\tau }\in \mathrm{PSH}(X,\theta )\cup \{ -\infty \} . \]
Due to proposition 17.12.16 and proposition 21.11.12, there is a pluripolar set $Z\subseteq X$ such that for $x\in X\setminus Z$, we have
\[ \begin{split} \psi _{\tau }(x)=\sup \left\{ \lambda \Gamma _{\rho }(x)+(1-\lambda )\Gamma '_{\rho '}(x): \lambda \in (0,1),\rho ,\rho '\in \mathbb {R},\lambda \rho +(1-\lambda )\rho '=\tau \right\} \end{split} \]
for all $\tau <\Gamma _{\max }\lor \Gamma '_{\max }$. It follows from proposition 5.14.6 that
1A2FC8833C52
17.15.16
\begin{equation} \begin{split} \psi _{\tau }=\operatorname *{sup}_{}{}\! {}^{*}\left\{ \lambda \Gamma _{\rho }+(1-\lambda )\Gamma '_{\rho '}: \lambda \in (0,1),\rho ,\rho '\in \mathbb {R},\lambda \rho +(1-\lambda )\rho '=\tau \right\} \end{split} \end{equation}
for all $\tau <\Gamma _{\max }\lor \Gamma '_{\max }$.
We now check that $\Gamma \lor \Gamma '$ is a test curve. The function $\rho \mapsto \Gamma _\rho \lor \Gamma '_\rho $ is decreasing, hence its upper concave envelope $\psi _\bullet $ is decreasing. The monotonicity of the $P$-envelope proposition 16.4.2 3 gives the decreasingness of $\Gamma \lor \Gamma '$. Since $\psi _\bullet $ is concave by construction, proposition 16.4.2 gives the concavity of $\Gamma \lor \Gamma '$.
Fix $\tau <\Gamma _{\max }\lor \Gamma '_{\max }$. Then either $\tau <\Gamma _{\max }$ or $\tau <\Gamma '_{\max }$. Since $\psi _\bullet $ is a concave majorant of $\rho \mapsto \Gamma _\rho \lor \Gamma '_\rho $, we have
\[ \psi _\tau \geq \Gamma _\tau \quad \text{if } \tau <\Gamma _{\max }, \qquad \psi _\tau \geq \Gamma '_\tau \quad \text{if } \tau <\Gamma '_{\max }. \]
Together with the fact that $\psi _\tau \in \mathrm{PSH}(X,\theta )\cup \{ -\infty \} $, these inequalities show that $\psi _\tau \in \mathrm{PSH}(X,\theta )$. It follows from 17.12.6 and the monotonicity theorem theorem 13.13.11 that $\psi _\tau \in \mathrm{PSH}(X,\theta )_{>0}$. By corollary 16.3.20,
\[ (\Gamma \lor \Gamma ')_\tau =P_\theta [\psi _\tau ] \]
is a model potential. Finally, $(\Gamma \lor \Gamma ')_{-\infty }\in \mathrm{PSH}(X,\theta )$ by proposition 5.14.2; since it dominates $(\Gamma \lor \Gamma ')_\tau $ for any $\tau <\Gamma _{\max }\lor \Gamma '_{\max }$, it has positive mass by another application of the monotonicity theorem theorem 13.13.11. Thus $\Gamma \lor \Gamma '\in \mathrm{TC}(X,\theta )_{>0}$.
It follows from (17.15.16) that $\psi _{\tau }$ is $\mathcal{I}$-good if $\Gamma ,\Gamma '\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$, thanks to proposition 18.1.17 and proposition 18.1.19; the family in the upper envelope is bounded above by $0$. Since $(\Gamma \lor \Gamma ')_\tau =P_\theta [\psi _\tau ]$, we conclude that $\Gamma \lor \Gamma '$ is $\mathcal{I}$-model in this case. $\square$
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.
\[ \bigvee _{i\in I}\Gamma _ i \]
without ambiguity.
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 $\Gamma ,\Gamma ',\Gamma ''$. $\square$
Lemma 17.15.19.5DB0F974F922 Let $\Gamma ,\Gamma '\in \mathrm{TC}(X,\theta )_{>0}$ and $\omega $ be a closed smooth positive $(1,1)$-form on $X$. Then
\[ P_{\theta +\omega }[\Gamma \lor \Gamma ']=P_{\theta +\omega }[\Gamma ]\lor P_{\theta +\omega }[\Gamma ']. \]
Proof. We first observe that
\[ \left(P_{\theta +\omega }[\Gamma \lor \Gamma ']\right)_{\max }=\left( P_{\theta +\omega }[\Gamma ]\lor P_{\theta +\omega }[\Gamma ']\right)_{\max }=\Gamma _{\max }\lor \Gamma '_{\max }. \]
Let $\tau \in \mathbb {R}$ be less than this common value. We need to show that
\[ (\Gamma \lor \Gamma ')_{\tau }\sim _ P \left(P_{\theta +\omega }[\Gamma ]\lor P_{\theta +\omega }[\Gamma '] \right)_{\tau }. \]
We need the formula (17.15.16) proved in the proof of 17.15.14:
\[ \begin{split} (\Gamma \lor \Gamma ')_{\tau }\sim _ P\operatorname *{sup}_{}{}\! {}^{*}\left\{ \lambda \Gamma _{\rho }+(1-\lambda )\Gamma '_{\rho '}: \lambda \in (0,1),\rho ,\rho '\in \mathbb {R},\lambda \rho +(1-\lambda )\rho '=\tau \right\} . \end{split} \]
A similar result holds with $P_{\theta +\omega }[\Gamma ]$ and $P_{\theta +\omega }[\Gamma ']$ in place of $\Gamma $ and $\Gamma '$. So our assertion is a direct consequence of proposition 16.8.27 and proposition 16.8.28. $\square$
Definition 17.15.20.4CC845CC4171 Let $(\Gamma ^ i)_{i\in I}$ be an increasing net in $\mathrm{TC}(X,\theta )_{>0}$. Assume that
Then we define $\operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i}\in \mathrm{TC}(X,\theta )_{>0}$ as follows:
E265E4ED4EFBWe set
\[ \left( \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right)_{\max }=\sup _{i\in I}\Gamma ^{i}_{\max }; \]A701AC5168C0for any $\tau < \sup _{i\in I}\Gamma ^{i}_{\max }$, we let
\[ \left( \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right)_{\tau }\coloneqq \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i}_{\tau }. \]
Proposition 17.15.22.88665EBBD399 Let $(\Gamma ^ i)_{i\in I}$ be an increasing net in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Then $\operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i}$ as defined in definition 17.15.20 lies in $\mathrm{TC}(X,\theta )_{>0}$. Moreover, if $\Gamma ^ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ for all $i\in I$, then $\operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i}$ lies in $\mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ as well.
Moreover, we have
75480BC1C778
17.15.23
\begin{equation} \left( \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right)_{-\infty }=\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i_{-\infty }. \end{equation}
Proof. Fix $\tau <\sup _ i\Gamma ^ i_{\max }$. Choose $i_0$ such that $\tau <\Gamma ^{i_0}_{\max }$. On the tail $i\geq i_0$, the potentials $\Gamma ^ i_\tau $ form an increasing net of model potentials of positive mass, uniformly bounded above by $0$. Hence proposition 16.4.9 shows that
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i_\tau = \operatorname *{sup}_{i\geq i_0}{}\! {}^{*}\Gamma ^ i_\tau \]
is model. Its mass is positive by monotonicity, since it dominates $\Gamma ^{i_0}_\tau $. The decreasingness and concavity in $\tau $ follow from the directedness of the net and the corresponding properties of the curves $\Gamma ^ i$. Thus $\operatorname *{sup}_{}{}\! {}^{*}_ i\Gamma ^ i$ is a test curve. If all $\Gamma ^ i$ are $\mathcal I$-model, the same tail argument and proposition 16.6.30 show that the supremum is $\mathcal I$-model.
It remains to argue (17.15.23). Fix any $i_0\in I$. Since the curves are decreasing and are extended as in 17.12.3, the value at $-\infty $ can be computed by taking the supremum over rational $\tau <\Gamma ^{i_0}_{\max }$.
By proposition 5.14.5, there is a pluripolar set $Z\subseteq X$ such that for any $x\in X\setminus Z$,
\[ \left( \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right)_{-\infty }(x)=\sup _{\mathbb {Q}\ni \tau < \Gamma ^{i_0}_{\max }}\left(\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i_{\tau }\right)(x)=\sup _{\mathbb {Q}\ni \tau < \Gamma ^{i_0}_{\max },i\in I}\Gamma ^ i_{\tau }(x)=\sup _{i\in I}\Gamma ^ i_{-\infty }(x). \]
So they are equal everywhere by proposition 5.14.6. $\square$
Lemma 17.15.24.0B89BAEA6D39 Let $(\Gamma ^ i)_{i\in I}$ be an increasing net in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Assume that $\omega $ is a closed smooth positive $(1,1)$-form on $X$. Then
\[ P_{\theta +\omega }\left[ \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right]= \operatorname *{sup}_{i\in I}{}\! {}^{*} P_{\theta +\omega }\left[ \Gamma ^{i} \right]. \]
Proof. Observe that
\[ \left( P_{\theta +\omega }\left[ \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right] \right)_{\max }= \left(\operatorname *{sup}_{i\in I}{}\! {}^{*} P_{\theta +\omega }\left[ \Gamma ^{i} \right]\right)_{\max }=\sup _{i\in I}\Gamma ^ i_{\max }. \]
Fix $\tau \in \mathbb {R}$ less than this common value.
It suffices to show that
\[ \left( \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right)_{\tau }\sim _ P \left( \operatorname *{sup}_{i\in I}{}\! {}^{*} P_{\theta +\omega }\left[\Gamma ^{i}\right] \right)_{\tau }. \]
This is an immediate consequence of proposition 16.8.28. $\square$
Definition 17.15.25.FA768FFB9E17 Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Let $\operatorname{Fin}(I)$ denote the net of non-empty finite subsets of $I$, ordered by inclusion. Then we define
80730711D47C
17.15.26
\begin{equation} \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i\coloneqq \operatorname *{sup}_{J\in \operatorname{Fin}(I)}{}\! {}^{*} \left( \bigvee _{j\in J} \Gamma ^ j\right). \end{equation}
Observe that by definition 17.15.12, we have
\[ \sup _{J\in \operatorname{Fin}(I)} \left( \bigvee _{j\in J} \Gamma ^ j\right)_{\max }=\sup _{i\in I}\Gamma ^ i_{\max }<\infty . \]
So (17.15.26) makes sense. In particular,
8A27FFEA5E96
17.15.27
\begin{equation} \left(\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i\right)_{\max }=\sup _{i\in I}\Gamma ^ i_{\max }. \end{equation}
It is clear that definition 17.15.25 extends both definition 17.15.20 and definition 17.15.12.
Proposition 17.15.28.C4A77DA880D9 Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Then $\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i\in \mathrm{TC}(X,\theta )_{>0}$. Moreover, if $\Gamma ^ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ for all $i\in I$, then so is $\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i$.
Finally, we have
3758587BAFAA
17.15.29
\begin{equation} \left( \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i \right)_{-\infty }=P_{\theta }\left[ \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i_{-\infty } \right]. \end{equation}
Proof. The first assertion and the second follow from proposition 17.15.22 and corollary 17.15.17.
It remains to argue (17.15.29). For this purpose, it suffices to show that
\[ \left( \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i \right)_{-\infty }\sim _ P \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i_{-\infty }. \]
For any $J\in \operatorname{Fin}(I)$, it follows from corollary 17.15.17 and proposition 16.8.28 that
\[ \left(\bigvee _{j\in J}\Gamma ^ j\right)_{-\infty }\sim _ P \bigvee _{j\in J}\Gamma ^ j_{-\infty }. \]
From this, applying proposition 16.4.9, proposition 16.8.28 and proposition 17.15.22, we conclude our assertion. $\square$
Lemma 17.15.30.B6D60981A394 Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Assume that $\omega $ is a closed smooth positive $(1,1)$-form on $X$. Then
\[ P_{\theta +\omega }\left[ \operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^{i} \right]= \operatorname *{sup}_{i\in I}{}\! {}^{*} P_{\theta +\omega }\left[ \Gamma ^{i} \right]. \]
Proof. This is a direct consequence of 17.15.24 and 17.15.19. $\square$
Proposition 17.15.31.4B67ABC0083D Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Then there is a countable subset $I'\subseteq I$ such that
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i =\operatorname *{sup}_{i\in I'}{}\! {}^{*}\Gamma ^ i. \]
Proof. We may assume that $I$ is infinite.
Set
\[ M\coloneqq \sup _{i\in I}\Gamma ^ i_{\max }. \]
For each rational number $q<M$, apply Choquet's lemma proposition 5.14.3 to the locally uniformly bounded above family
\[ \left\{ \left(\bigvee _{j\in J}\Gamma ^ j\right)_ q:J\in \operatorname{Fin}(I)\right\} \subseteq \mathrm{PSH}(X,\theta ). \]
We obtain a countable subfamily $\mathcal J_ q\subseteq \operatorname{Fin}(I)$ such that
\[ \operatorname *{sup}_{J\in \operatorname{Fin}(I)}{}\! {}^{*}\left(\bigvee _{j\in J}\Gamma ^ j\right)_ q = \operatorname *{sup}_{J\in \mathcal J_ q}{}\! {}^{*}\left(\bigvee _{j\in J}\Gamma ^ j\right)_ q. \]
Let
\[ I'\coloneqq \bigcup _{q\in \mathbb Q, q<M} \bigcup _{J\in \mathcal J_ q}J. \]
Then $I'$ is countable and, for every rational $q<M$,
\[ \left(\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i\right)_ q= \left(\operatorname *{sup}_{i\in I'}{}\! {}^{*}\Gamma ^ i\right)_ q. \]
Moreover, $\sup _{i\in I'}\Gamma ^ i_{\max }=M$, since the last equality is non-trivial for every rational $q<M$. Let $\Gamma '=\operatorname *{sup}_{i\in I'}{}\! {}^{*}\Gamma ^ i$. Then clearly, $\Gamma '\leq \Gamma $. We claim that they are actually equal.
Thanks to proposition 16.8.7 and 17.12.6, it suffices to show that for any $\tau <\operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i_{\max }$, we have
\[ \int _ X \left(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma '_{\tau }\right)^ n=\int _ X \left(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma _{\tau }\right)^ n. \]
The equality is already known for all rational $\tau <M$. Since both
\[ \tau \mapsto \log \int _ X\theta _{\Gamma '_\tau }^ n,\quad \tau \mapsto \log \int _ X\theta _{\Gamma _\tau }^ n \]
are continuous on $(-\infty ,M)$ by proposition 17.12.9, the equality of masses holds for every $\tau <M$. $\square$
Proposition 17.15.32.4D160D08BDC8 Let $(\Gamma ^ i)_{i\in I}$ be a non-empty family in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Let $C\in \mathbb {R}$. Then
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*} (\Gamma ^ i+C)=\operatorname *{sup}_{i\in I}{}\! {}^{*} \Gamma ^ i+C. \]
Suppose that $(\Gamma '^ i)_{i\in I}$ is another family in $\mathrm{TC}(X,\theta ')_{>0}$ satisfying (17.15.21). Suppose that $\Gamma ^ i\leq \Gamma '^ i$ for all $i\in I$. Then
\[ \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i\leq \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma '^ i. \]
Proof. This is immediate by definition. $\square$
Definition 17.15.33.F6690BE17BAF Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$ and $\lambda >0$, we define $\lambda \Gamma \in \mathrm{TC}(X,\lambda \theta )_{>0}$ as follows:
1297F6EE2D1DWe set
\[ \left(\lambda \Gamma \right)_{\max }=\lambda \Gamma _{\max }; \]4310047CFEDDfor any $\tau < \left(\lambda \Gamma \right)_{\max }$, we set
\[ (\lambda \Gamma )_{\tau }=\lambda \Gamma _{\lambda ^{-1}\tau }. \]
Proposition 17.15.34.148305D820BC Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$ and $\lambda >0$. Then $\lambda \Gamma $ as defined in definition 17.15.33 lies in $\mathrm{TC}(X,\lambda \theta )_{>0}$. Moreover, if $\Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$, then $\lambda \Gamma \in \mathrm{PSH}^{\mathrm{NA}}(X,\lambda \theta )_{>0}$.
We have
9.92
\begin{equation} (\lambda \Gamma )_{-\infty }=\lambda \Gamma _{-\infty }. \end{equation}
Proof. This is immediate by definition. $\square$
Proposition 17.15.35.A44C0F3CE359 Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$, $\Gamma '\in \mathrm{TC}(X,\theta ')_{>0}$, $C\in \mathbb {R}$ and $\lambda ,\lambda '>0$, we have
\[ \begin{aligned} \lambda (\Gamma +\Gamma ')=& \lambda \Gamma +\lambda \Gamma ',\\ (\lambda \lambda ')\Gamma =& \lambda (\lambda '\Gamma ),\\ \lambda (\Gamma +C)=& \lambda \Gamma +\lambda C. \end{aligned} \]
Suppose that $(\Gamma ^ i)_{i\in I}$ is a non-empty family in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.21). Then
\[ \lambda \left( \operatorname *{sup}_{i\in I}{}\! {}^{*}\Gamma ^ i \right)=\operatorname *{sup}_{i\in I}{}\! {}^{*}(\lambda \Gamma ^ i). \]
Proof. This is immediate by definition. $\square$
Lemma 17.15.36.9469C1C052AA Let $\Gamma \in \mathrm{TC}(X,\theta )_{>0}$ and $\lambda >0$. Then for any closed smooth positive $(1,1)$-form $\omega $ on $X$, we have
\[ P_{\lambda \theta +\lambda \omega }[\lambda \Gamma ]=\lambda P_{\theta +\omega }[\Gamma ]. \]
Proof. This is clear by definition. $\square$
Definition 17.15.37.BDCF69FED30F Let $(\Gamma ^ i)_{i\in I}$ be a decreasing net in $\mathrm{TC}(X,\theta )_{>0}$. Assume that
and
C2C47DDCADA4
17.15.39
\begin{equation} \inf _{i\in I} \int _ X \left( \theta +\mathrm{dd}^{\mathrm{c}}\Gamma ^ i_{\tau } \right)^ n>0,\quad \textup{for some }\tau < \inf _{i\in I}\Gamma ^{i}_{\max }. \end{equation}
Then we define $\inf _{i\in I} \Gamma ^{i}\in \mathrm{TC}(X,\theta )_{>0}$ as follows:
3E567FA790CCWe set
\[ \left( \inf _{i\in I} \Gamma ^{i} \right)_{\max }=\inf _{i\in I}\Gamma ^{i}_{\max }; \]88EFD81042D1for any $\tau < \left( \inf _{i\in I} \Gamma ^{i} \right)_{\max }$, we let
\[ \left( \inf _{i\in I} \Gamma ^{i} \right)_{\tau }\coloneqq \inf _{i\in I} \Gamma ^{i}_{\tau }. \]
Proposition 17.15.40.82DA4D86CC78 Let $(\Gamma ^ i)_{i\in I}$ be a decreasing net in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.38) and (17.15.39). Then $\inf _{i\in I} \Gamma ^{i}\in \mathrm{TC}(X,\theta )_{>0}$.
Moreover, if $\Gamma ^ i\in \mathrm{PSH}^{\mathrm{NA}}(X,\theta )_{>0}$ for all $i\in I$, then so is $\inf _{i\in I} \Gamma ^{i}$.
Proof. The first assertion is an immediate consequence of proposition 16.4.3, except for the positivity of the mass at $-\infty $. Choose $\tau _0$ as in (17.15.39). Applying proposition 16.4.4 gives
\[ \int _ X \left(\theta +\mathrm{dd}^{\mathrm{c}}\inf _{i\in I}\Gamma ^ i_{\tau _0}\right)^ n =\lim _{i\in I}\int _ X\left(\theta +\mathrm{dd}^{\mathrm{c}}\Gamma ^{i}_{\tau _0}\right)^ n>0. \]
Hence the value at $-\infty $ has positive mass by monotonicity. The last assertion follows from proposition 16.6.29. $\square$
In general, it is not true that
\[ \left( \inf _{i\in I}\Gamma ^{i} \right)_{-\infty }=\inf _{i\in I}\Gamma ^{i}_{-\infty }. \]
Lemma 17.15.41.E141137C118E Let $(\Gamma ^ i)_{i\in I}$ be a decreasing net in $\mathrm{TC}(X,\theta )_{>0}$ satisfying (17.15.38) and (17.15.39). Assume that $\omega $ is a closed smooth positive $(1,1)$-form on $X$. Then
\[ P_{\theta +\omega }\left[ \inf _{i\in I} \Gamma ^{i} \right]= \inf _{i\in I} P_{\theta +\omega }\left[ \Gamma ^{i} \right]. \]
Proof. First observe that
\[ \left( P_{\theta +\omega }\left[ \inf _{i\in I} \Gamma ^{i} \right] \right)_{\max }= \left(\inf _{i\in I} P_{\theta +\omega }\left[ \Gamma ^{i} \right]\right)_{\max }. \]
Let $\tau \in \mathbb {R}$ be less than this common value. Then we need to show the following:
1D25957452B7
17.15.42
\begin{equation} P_{\theta +\omega }\left[ \inf _{i\in I} \Gamma ^ i_{\tau } \right]\sim _ P \inf _{i\in I} P_{\theta +\omega }\left[ \Gamma ^ i_{\tau } \right]. \end{equation}
Applying proposition 16.4.4 gives the mass convergence required in corollary 16.9.14. Hence $\Gamma ^ i_{\tau }\xrightarrow {d_ S}\inf _{j\in I} \Gamma ^ j_{\tau } $. Thanks to corollary 16.9.36 and corollary 16.9.14, we have
\[ P_{\theta +\omega }\left[ \Gamma ^ i_{\tau } \right]\xrightarrow {d_ S} P_{\theta +\omega }\left[ \inf _{j\in I} \Gamma ^ j_{\tau } \right],\quad P_{\theta +\omega }\left[ \Gamma ^ i_{\tau } \right]\xrightarrow {d_ S} \inf _{j\in I} P_{\theta +\omega }\left[ \Gamma ^ j_{\tau } \right]. \]
Hence, (17.15.42) follows from proposition 16.9.6. $\square$
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