Definition 16.29.2F0A985779829

Definition 16.29.2.F0A985779829 Let $\mathscr T_{a,c}(\theta )$ be the set of families $\Gamma =(\Gamma _\tau )_{\tau <c}$ such that

1. $\Gamma _\tau \in \mathrm{PSH}(X,\theta )$ and $P_\theta [\Gamma _\tau ]_{\mathcal I}=\Gamma _\tau $ for every $\tau <c$;

2. the family is pointwise concave and decreasing in $\tau $;

3. $\Gamma _\tau =\Gamma _ a$ for $\tau \leq a$.

The cutoff of $\Gamma $ is $c$, and we extend $\Gamma $ by $\Gamma _ c=\lim _{\tau \uparrow c}\Gamma _\tau $ and $\Gamma _\tau =-\infty $ for $\tau >c$. Such curves are ordered by the pointwise order of their slices $\Gamma _\tau $, $\tau <c$.