Corollary 17.10.9.C645EE3517E1 Let $\varphi _0,\varphi _1\in \mathrm{PSH}(X,\theta )$ and $(\varphi _ t)_{t\in (0,1)}$ be a subgeodesic from $\varphi _0$ to $\varphi _1$. Then
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17.10.10
\begin{equation} \lim _{t\to 0+}\varphi _ t\leq \varphi _0,\quad \lim _{t\to 1-}\varphi _ t\leq \varphi _1. \end{equation}
Moreover,
\[ \lim _{t\to 0+}\varphi _ t=\varphi _0,\quad \lim _{t\to 1-}\varphi _ t=\varphi _1\quad \textup{quasi-everywhere}. \]
Proof.
The inequalities in (17.10.10) and the existence of the limits follow from the convexity in proposition 17.10.7. Let us prove the quasi-everywhere equality at $t=0$. Take a decreasing sequence $t_ j\searrow 0$ with $t_ j>0$. Then $\varphi _{t_ j}\to \varphi _0$ in $L^1(X)$ as $j\to \infty $. So by corollary 5.14.8, we have
\[ \varlimsup _{j\to \infty }\varphi _{t_ j}= \varphi _0\quad \textup{quasi-everywhere}. \]
Therefore, we have
\[ \varphi _0=\varlimsup _{j\to \infty }\varphi _{t_ j}=\lim _{t\to 0+}\varphi _ t\leq \varphi _0\quad \textup{quasi-everywhere}. \]
The assertion at $t=1$ is similar.
$\square$
Pome
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