PomeLemma 17.13.15.7B83BD1D5FB9 Given $\ell \in \mathcal{R}(X,\theta ;\phi )$, then $\ell ^*=(\ell ^*_\tau )_{\tau <\sup _ X \ell _1}\in \mathrm{TC}(X,\theta ;\phi )$.
Lemma 17.13.157B83BD1D5FB9
Proof. Note that it follows from proposition 5.14.15 that $\ell ^*_{\tau }\in \mathrm{PSH}(X,\theta )\cup \{ -\infty \} $ for all $\tau \in \mathbb {R}$. It is clear that $\mathbb {R}\ni \tau \mapsto \ell ^*_{\tau }$ is a decreasing and concave function.
By proposition 17.10.39,
\[ \sup _ X\ell _ t= t\sup _ X \ell _1 \quad \forall t\geq 0. \]
Set $C=\sup _ X\ell _1$. By proposition 17.10.39, $\sup _ X(\ell _ t-Ct)=0$ for all $t>0$. For each $x$ outside a pluripolar set, the function $t\mapsto \ell _ t(x)-Ct$ is convex on $(0,\infty )$ and bounded above by $0$, hence it is decreasing. By proposition 5.14.6, the same monotonicity holds as an inequality of $\theta $-psh functions. Thus $(\ell _ t-Ct)_{t>0}$ is a decreasing net in $\mathrm{PSH}(X,\theta )$.
Its infimum is not identically $-\infty $, since the family is normalized by $\sup _ X(\ell _ t-Ct)=0$ and is compact in $L^1$ by proposition 5.17.3. Therefore
\[ \ell ^*_{C}=\inf _{t>0}(\ell _ t-Ct)\in \mathrm{PSH}(X,\theta ) \]
by proposition 5.14.2.
On the other hand, for $\tau >\sup _ X \ell _1$, the same argument shows that
\[ \ell ^*_{\tau }\equiv -\infty . \]
Therefore, $\ell ^*_{\tau }\in \mathrm{PSH}(X,\theta )$ if and only if $\tau \leq \ell ^*_{\max }\coloneqq \sup _ X \ell _1$.
We claim that $(\ell ^*_{\tau })_{\tau <\ell ^*_{\max }}$ is a test curve. We first observe that for $\tau <\ell ^*_{\max }$, we have
\[ \ell ^*_{\tau }\leq \ell _1-\tau \sim _ P \phi . \]
Therefore,
5BF1777D23C2
17.13.16
\begin{equation} \ell ^*_{\tau }\preceq _ P \phi ,\quad \forall \tau <\ell ^*_{\max }. \end{equation}
Also observe that for any $\tau \leq \ell ^*_{\max }$ and any $t>0$, we have
\[ \sup _ X \ell ^*_{\tau }\leq \sup _ X \ell _ t-t\tau =\ell ^*_{\max }t-t\tau . \]
Letting $t\to 0+$, we find that for any $\tau \leq \ell ^*_{\max }$, we have
Fix $\tau <\ell ^*_{\max }$. We want to argue that
59C9610616F1
17.13.18
\begin{equation} P_{\theta }\left[\ell ^*_{\tau }\right]=\ell ^*_{\tau }. \end{equation}
First we claim that for any $C>0$, we have
25F16D030FA9
17.13.19
\begin{equation} (\ell ^*_{\tau }+C)\land \phi =(\ell ^*_{\tau }+C)\land V_{\theta }. \end{equation}
The $\leq $ direction is trivial. We argue the reverse inequality, which reduces to
\[ \phi \geq (\ell ^*_{\tau }+C)\land V_{\theta }. \]
Since $\phi $ is model and $(\ell ^*_{\tau }+C)\land V_{\theta }\leq 0$, it suffices to show that
\[ \phi \succeq _ P (\ell ^*_{\tau }+C)\land V_{\theta }, \]
which follows from (17.13.16). Therefore, (17.13.19) is established. Thanks to (17.13.17), we have the obvious inequality
\[ (\ell ^*_{\tau }+C)\land V_{\theta }\geq \ell ^*_{\tau } \]
for any $C>0$. Therefore, in order to prove (17.13.18), it remains to argue that for any $C>0$,
CBA3E07AB082
17.13.20
\begin{equation} (\ell ^*_{\tau }+C)\land \phi \leq \ell ^*_{\tau }. \end{equation}
For this purpose, let us consider the following geodesics: For any $M>0$ and $t\in [0,1]$, let
\[ \ell ^{1,M}_ t= \ell _{tM}-tM\tau ,\quad \ell ^{2,M}_ t= (\ell ^*_{\tau }+C)\land \phi -Ct. \]
It is clear that at $t=0,1$, we have $\ell ^{2,M}_ t\leq \ell ^{1,M}_ t$. Hence, the same holds for all $t\in [0,1]$. In particular, for any fixed $s\in (0,1]$, we have
\[ (\ell ^*_{\tau }+C)\land \phi -Cs\leq \ell _{sM}-sM\tau \]
for all $M>0$. Taking infimum with respect to $M>0$, we find
\[ (\ell ^*_{\tau }+C)\land \phi -Cs\leq \ell ^*_{\tau }. \]
Since $s\in (0,1]$ is arbitrary, we conclude (17.13.20).
Finally, we check the condition at $-\infty $. Set
\[ \psi \coloneqq \operatorname *{sup}_{\tau <\ell ^*_{\max }}{}\! {}^{*}\ell ^*_{\tau }. \]
The family $(\ell ^*_{\tau })_{\tau <\ell ^*_{\max }}$ is locally uniformly bounded from above, so $\psi \in \mathrm{PSH}(X,\theta )$. Thanks to corollary 17.10.9, outside a pluripolar set, the function $t\mapsto \ell _ t(x)$ is a finite closed convex function on $[0,\infty )$ with value $\phi (x)$ at $t=0$. Applying the one-variable Legendre duality gives
\[ \psi (x)=\adjustlimits \sup _{\tau <\ell ^*_{\max }}\inf _{t>0}\bigl (\ell _ t(x)-t\tau \bigr )=\ell _0(x)=\phi (x)\quad \textup{quasi-everywhere}. \]
Hence $\psi =\phi $ by proposition 5.14.6. In particular, $\ell ^*_{-\infty }=\phi $ has positive mass. $\square$
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