Corollary 13.13.12.37623491DEDC Fix a directed set $I$. For each $j=1,\ldots ,n$, take an increasing net $(\varphi _ j^ i)_{i\in I}$ in $\mathrm{PSH}(X,\theta _ j)$, uniformly bounded from above. Set
\[ \varphi _ j\coloneqq \operatorname *{sup}_{i\in I}{}\! {}^{*} \varphi _ j^ i,\quad j=1,\ldots ,n. \]
Then
18A95DBDA47A
13.13.13
\begin{equation} \lim _{i\in I}\int _ X \theta _{1,\varphi _1^ i}\wedge \cdots \wedge \theta _{n,\varphi _ n^ i} = \int _ X \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n}. \end{equation}
In particular, if the nets are indeed sequences, we have
\[ \theta _{1,\varphi _1^ i}\wedge \cdots \wedge \theta _{n,\varphi _ n^ i} \rightharpoonup \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n}. \]
Proof.
The final assertion follows from (13.13.13) together with theorem 13.13.9. It remains to establish (13.13.13).
We may assume that $I$ is infinite as there is nothing to prove otherwise. Thanks to theorem 13.13.11, we already know the $\leq $ inequality in (13.13.13). We prove the reverse inequality. When $I\cong \mathbb {Z}_{>0}$ as directed sets, the reverse inequality follows from theorem 13.13.9. In general, by Choquet's lemma proposition 5.14.3, we can find a countable infinite subset $R\subseteq I$ such that
\[ \operatorname *{sup}_{r\in R}{}\! {}^{*}\varphi ^ r_ j=\operatorname *{sup}_{i\in I}{}\! {}^{*}\varphi ^ i_ j \]
for all $j=1,\ldots ,n$. We fix a bijection $R\cong \mathbb {Z}_{>0}$. For any $j=1,\ldots ,n$, we will then denote elements $\varphi _ j^ r$ ($r\in R$) by $\varphi _ j^1,\varphi _ j^2,\dots $. We shall write
\[ \psi _ j^ a=\varphi _ j^1\lor \cdots \lor \varphi _ j^ a \]
for each $a\in \mathbb {Z}_{>0}$.
For every $a$ we may choose $i_ a\in I$ dominating the finitely many indices corresponding to $\varphi _ j^1,\ldots ,\varphi _ j^ a$ for all $j$. It then follows from the monotonicity theorem that
\[ \lim _{i\in I}\int _ X \theta _{1,\varphi _1^ i}\wedge \cdots \wedge \theta _{n,\varphi _ n^ i}\geq \lim _{a\to \infty }\int _ X \theta _{1,\psi _1^ a}\wedge \cdots \wedge \theta _{n,\psi _ n^ a}. \]
From the special case mentioned above, we know that the right-hand side is exactly the right-hand side of (13.13.13), so we conclude.
$\square$
Pome
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