Theorem 13.12.8272876B00528

Theorem 13.12.8.272876B00528 Let $U$ be an open subset of $X$. Let $(\chi _ i\colon U\rightarrow [0,\infty ))_ i$ be a sequence of quasi-continuous functions. Assume that the sequence is uniformly bounded and converges in capacity to $\chi \colon U\rightarrow [0,\infty )$. For each $j=1,\ldots ,p$, let $(\varphi _ j^ i)_ i$ be uniformly bounded sequences in $\mathrm{PSH}(U,\theta _ j)$ converging in capacity to bounded $\varphi _ j\in \mathrm{PSH}(U,\theta _ j)$. Then

weakly as currents on $U$.