Proposition 17.10.77.7ADCF3A953E5 Let $(\varphi ^ i_0)_{i\in I}$, $(\varphi ^ i_1)_{i\in I}$ be two uniformly bounded from above increasing nets in $\mathcal{E}^{1}(X,\theta ;\phi )$. Let $(\varphi ^ i_ t)_{t\in (0,1)}$ be the geodesic from $\varphi _0^ i$ to $\varphi _1^ i$ for each $i\in I$. Then
\[ \left(\operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi ^ i_ t\right)_{t\in (0,1)} \]
is the geodesic from $\operatorname *{sup}_{i}{}\! {}^{*} \varphi ^ i_0$ to $\operatorname *{sup}_{i}{}\! {}^{*} \varphi ^ i_1$.
Proof.
Put
\[ \varphi _ j=\operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi _ j^ i\quad (j=0,1), \qquad h_ t=\operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi _ t^ i, \]
and let $(g_ t)_{t\in (0,1)}$ be the geodesic from $\varphi _0$ to $\varphi _1$. By proposition 16.5.53, $\varphi _0,\varphi _1\in \mathcal E^1(X,\theta ;\phi )$. The slice families are uniformly bounded from above by proposition 17.10.7; hence $h_ t\in \mathcal E^1(X,\theta ;\phi )$ by proposition 16.5.53. Moreover, proposition 17.10.13 shows that $(h_ t)$ is a subgeodesic from $\varphi _0$ to $\varphi _1$. Thus
D9CCCF0D8552
17.10.78
\begin{equation} h_ t\leq g_ t. \end{equation}
For each $t$, the net $(\varphi _ t^ i)_{i\in I}$ is increasing. By proposition 5.14.5, its pointwise supremum agrees with $h_ t$ almost everywhere, and the analogous assertion holds at the endpoints. Therefore proposition 16.5.37, proposition 17.10.65 gives
\[ \begin{aligned} E_\theta ^\phi (h_ t) & =\lim _{i\in I}E_\theta ^\phi (\varphi _ t^ i)\\ & =\lim _{i\in I}\left((1-t)E_\theta ^\phi (\varphi _0^ i) +tE_\theta ^\phi (\varphi _1^ i)\right)\\ & =(1-t)E_\theta ^\phi (\varphi _0)+tE_\theta ^\phi (\varphi _1) =E_\theta ^\phi (g_ t). \end{aligned} \]
By (17.10.42), (17.10.78), and 17.10.45,
\[ d_1(h_ t,g_ t)=E_\theta ^\phi (g_ t)-E_\theta ^\phi (h_ t)=0, \]
so $h_ t=g_ t$ for all $t\in (0,1)$.
$\square$
Pome
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