PomeLemma 19.5.16.08C830D7FB38 The convex body $\Delta _{Y_{\bullet }}(T)$ in definition 19.5.15 is independent of the choices of the $T_ j$'s.
Lemma 19.5.1608C830D7FB38
Proof. It is more convenient to use the language of $\theta $-psh functions at this point. Let $\psi _ k$ (resp. $\varphi _ k$) denote the potentials in $\mathrm{PSH}(X,\theta )$ corresponding to $S_ k$ (resp. $T_ k$) for each $k\geq 1$. Note that $\psi _ k$ and $\varphi _ k$ are unique up to additive constants. By proposition 5.18.12, for any small rational $\epsilon >0$, $j>0$, we can find $k>0$ so that
\[ \psi _ k\preceq (1-\epsilon )\varphi _ j. \]
By 19.5.12,
\[ \bigcap _{k=1}^{\infty }\Delta _{Y_{\bullet }}(\theta ,\psi _ k)\subseteq \Delta _{Y_{\bullet }}(\theta ,(1-\epsilon )\varphi _ j). \]
On the other hand, observe that
\[ \bigcap _{\epsilon \in \mathbb {Q}_{>0}\text{ small enough}}\Delta _{Y_{\bullet }}(\theta ,(1-\epsilon )\varphi _ j)=\Delta _{Y_{\bullet }}(\theta ,\varphi _ j). \]
In fact, the $\supseteq $ direction follows from 19.5.12, so it suffices to show that the two sides have the same volume, which follows from (19.5.13).
It follows that
\[ \bigcap _{k=1}^{\infty }\Delta _{Y_{\bullet }}(\theta ,\psi _ k)\subseteq \bigcap _{j=1}^{\infty }\Delta _{Y_{\bullet }}(\theta ,\varphi _ j). \]
The other inclusion follows by symmetry. $\square$
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