Proposition 16.6.29.2D7AF550FFF7 Consider a decreasing net $(\varphi _ i)_{i\in I}$ of model potentials, with $\varphi _ i\in \mathrm{PSH}(X,\theta )_{>0}$ for all $i\in I$. Set
\[ \varphi \coloneqq \inf _{i\in I}\varphi _ i. \]
Suppose that $\varphi \not\equiv -\infty $ and $\int _ X \theta _{\varphi }^ n>0$. Then
\[ \inf _{i\in I} P_{\theta }[\varphi _ i]_{\mathcal{I}}=P_{\theta }[\varphi ]_{\mathcal{I}}. \]
In particular, if the $\varphi _ i$'s are all $\mathcal{I}$-model, then so is $\varphi $.
Proof.
Let $\eta =\inf _{i\in I} P_{\theta }[\varphi _ i]_{\mathcal{I}}$. We have $\eta \geq P_{\theta }[\varphi ]_{\mathcal{I}}$ as a consequence of 16.6.28. In particular, $\eta \in \mathrm{PSH}(X,\theta )$.
By proposition 16.4.4, we have
\[ \lim _{i\in I} \int _ X\theta _{\varphi _ i}^ n=\int _ X\theta _{\varphi }^ n>0. \]
We may assume that for each $i\in I$, $\int _ X\theta _{\varphi _ i}^ n>\int _ X\theta _{\varphi }^ n$, since otherwise $\varphi _ i=\varphi $ by theorem 16.3.15, and hence $P_{\theta }[\varphi _ i]_{\mathcal{I}}=P_{\theta }[\varphi ]_{\mathcal{I}}$ due to (16.6.19); the same holds for all $j\geq i$, and there is nothing to prove.
So by 13.13.30, we can find a decreasing net $\epsilon _ i\searrow 0$ ($i\in I$) with $\epsilon _ i\in (0,1)$ and $\psi _ i\in \mathrm{PSH}(X,\theta )$ such that for all $i\in I$,
\[ (1-\epsilon _ i) \varphi _ i+\epsilon _ i\psi _ i\leq \varphi . \]
We can, for example, take
\[ \epsilon _ i^{-1}=\frac{1}{2}\left( 1+\left(\frac{\int _ X \theta _{\varphi _ i}^ n}{\int _ X \theta _{\varphi _ i}^ n-\int _ X \theta _{\varphi }^ n}\right)^{1/n} \right). \]
By proposition 16.6.27 and 16.6.28, we have
\[ \eta +\epsilon _ i P_{\theta }[\psi _ i]_{\mathcal{I}}\leq (1-\epsilon _ i) \eta +\epsilon _ i P_{\theta }[\psi _ i]_{\mathcal{I}}\leq (1-\epsilon _ i) P_{\theta }[\varphi _ i]_{\mathcal{I}}+\epsilon _ i P_{\theta }[\psi _ i]_{\mathcal{I}}\leq P_{\theta }[\varphi ]_{\mathcal{I}}. \]
Each $P_{\theta }[\psi _ i]_{\mathcal{I}}$ has supremum $0$, so proposition 5.17.3 shows that the family is bounded in $L^1$. Hence $\epsilon _ iP_{\theta }[\psi _ i]_{\mathcal{I}}\to 0$ in $L^1$, and the preceding inequality gives $\eta \leq P_{\theta }[\varphi ]_{\mathcal{I}}$ almost everywhere. By proposition 5.14.6, the inequality holds everywhere.
$\square$
Pome
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