Proof.
We first show that for each fixed $t\geq 0$, $\ell _ t^{\varphi ,C}$ is increasing in $C\geq t$.
To see this, choose $t\leq C_1<C_2$. We need to show that
\[ \ell ^{\varphi ,C_1}_ t\leq \ell ^{\varphi ,C_2}_ t. \]
Since both sides are geodesics for $t\in [0,C_1]$, it suffices to show that
277396283395
17.10.95
\begin{equation} (V_{\theta }-C_1)\lor \varphi \leq \ell _{C_1}^{\varphi ,C_2}. \end{equation}
Now $((V_{\theta }-t)\lor \varphi )_{t\in [0,C_2]}$ is a subgeodesic from $V_{\theta }$ to $(V_{\theta }-C_2)\lor \varphi $ by proposition 17.10.13. It is therefore a candidate in the Perron envelope defining $\ell ^{\varphi ,C_2}$, and evaluating at $t=C_1$ gives exactly (17.10.95).
From proposition 17.10.7, we know that for any $C>t>0$, we have
\[ \ell ^{\varphi ,C}_ t\leq \frac{t}{C}\Bigl ((V_{\theta }-C)\lor \varphi \Bigr )+\frac{C-t}{C}\cdot V_{\theta }\leq 0, \]
while the linear subgeodesic $V_\theta -t$ is a candidate in the same Perron envelope, so $V_\theta -t\leq \ell ^{\varphi ,C}_ t$. Therefore, by proposition 5.14.2,
AAACF2ED8350
17.10.96
\begin{equation} \ell _ t^{\varphi }\coloneqq \operatorname *{sup}_{C>t}{}\! {}^{*}\ell ^{\varphi ,C}_ t\in \mathcal{E}^{\infty }(X,\theta ) \end{equation}
for all $t\geq 0$. Thanks to proposition 16.5.37, we have
\[ \ell ^{\varphi ,C}_ t\xrightarrow {d_1} \ell _ t^{\varphi } \]
as $C\to \infty $ for all $t\geq 0$. Applying proposition 17.10.77 on each interval $[0,T]$ to the increasing family $(\ell ^{\varphi ,C}_ t)_{t\in [0,T]}$, $C\geq T$, shows that $\ell ^{\varphi }\in \mathcal{R}^{\infty }(X,\theta )$.
It remains to compute the energy of $\ell ^{\varphi }$. We first fix $C\geq t>0$ and compute using proposition 17.10.65:
\[ E_{\theta }\left(\ell ^{\varphi ,C}_ t\right)=\frac{t}{C}E_{\theta }\Bigl ((V_{\theta }-C)\lor \varphi \Bigr ). \]
Letting $C\to \infty $ and applying proposition 16.5.37, we find that
\[ E_{\theta }(\ell ^{\varphi }_ t)=\lim _{C\to \infty }\frac{t}{C}E_{\theta }\Bigl ((V_{\theta }-C)\lor \varphi \Bigr ) \]
for any $t\geq 0$. It follows that
\[ \mathbf{E}(\ell ^{\varphi })=\lim _{C\to \infty }\frac{1}{C}E_{\theta }\Bigl ((V_{\theta }-C)\lor \varphi \Bigr ). \]
Using the definition of $E_{\theta }$, in order to obtain (17.10.94), it suffices to show that for each $j=0,\ldots ,n$, we have
287169AF175C
17.10.97
\begin{equation} \lim _{C\to \infty }\int _ X \frac{(V_{\theta }-C)\lor \varphi -V_{\theta }}{C} \, \theta _{(V_{\theta }-C)\lor \varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}=\int _ X \theta _{\varphi }^{j}\wedge \theta _{V_{\theta }}^{n-j}-\int _ X \theta _{V_{\theta }}^ n. \end{equation}
For this purpose, for each $C>0$, we decompose $X$ as $\{ \varphi > V_{\theta }-C\} $ and $\{ \varphi \leq V_{\theta }-C\} $. We have
\[ \begin{aligned} & \int _{\{ \varphi > V_{\theta }-C\} } \frac{(V_{\theta }-C)\lor \varphi -V_{\theta }}{C} \, \theta _{(V_{\theta }-C)\lor \varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}\\ =& \int _{\{ \varphi > V_{\theta }-C\} } \frac{\varphi -V_{\theta }}{C} \, \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}. \end{aligned} \]
On the other hand,
\[ \begin{aligned} & \int _{\{ \varphi \leq V_{\theta }-C\} } \frac{(V_{\theta }-C)\lor \varphi -V_{\theta }}{C} \, \theta _{(V_{\theta }-C)\lor \varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}\\ =& -\int _{\{ \varphi \leq V_{\theta }-C\} } \theta _{(V_{\theta }-C)\lor \varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}\\ =& -\int _ X \theta _{V_{\theta }}^ n+\int _{\{ \varphi > V_{\theta }-C\} } \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}. \end{aligned} \]
Observe that for $C>0$, the function $\mathds {1}_{\{ \varphi >V_{\theta }-C\} }C^{-1}(\varphi -V_{\theta })$ is defined quasi-everywhere. Since $\varphi \leq V_\theta $, it takes values in $[-1,0]$ outside a pluripolar set. When $C\to \infty $, these functions converge to $0$ quasi-everywhere, so the first integral above converges to $0$ by dominated convergence. Similarly, the indicators $\mathds {1}_{\{ \varphi >V_\theta -C\} }$ increase to $1$ outside a pluripolar set, hence
\[ \lim _{C\to \infty }\int _{\{ \varphi >V_{\theta }-C\} } \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j} = \int _ X \theta _{\varphi }^ j\wedge \theta _{V_{\theta }}^{n-j}. \]
Therefore, (17.10.97) follows.
$\square$
Pome
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