PomeTheorem 13.13.9 (Semicontinuity theorem).B3DC3180FF35 Let $\varphi _ j, \varphi _ j^ k\in \mathrm{PSH}(X,\theta _ j)$ ($k\in \mathbb {N}$, $j=1,\ldots ,n$). Let $\chi _ k,\chi $ ($k\in \mathbb {N}$) be non-negative uniformly bounded quasi-continuous functions on $X$ such that $(\chi _ k)_ k$ converges to $\chi $ in capacity. Assume that for any $j=1,\ldots ,n$, as $k\to \infty $, either $\varphi _ j^ k$ decreases to $\varphi _ j\in \mathrm{PSH}(X,\theta _ j)$ or increases to $\varphi _ j\in \mathrm{PSH}(X,\theta _ j)$ almost everywhere. Then
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13.13.10
\begin{equation} \varliminf _{k\to \infty }\int _ X\chi _ k \, \theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k} \geq \int _ X \chi \, \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n}. \end{equation}
If in addition,
\[ \lim _{k\to \infty }\int _ X\theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k} =\int _ X \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n}, \]
then
\[ \chi _ k \, \theta _{1,\varphi _1^ k}\wedge \cdots \wedge \theta _{n,\varphi _ n^ k}\rightharpoonup \chi \, \theta _{1,\varphi _1}\wedge \cdots \wedge \theta _{n,\varphi _ n} \]
weakly as measures3. In particular, the limit in (13.13.10) exists and equality holds in (13.13.10).
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