Proposition 13.12.7D56D6D4A5E94

Proposition 13.12.7.D56D6D4A5E94 Let $U$ be an open subset of $X$. Let $(\varphi _ i)_{i\in I}$ be a net in $\mathrm{PSH}(U,\theta )$ and $\varphi \in \mathrm{PSH}(U,\theta )$. Assume one of the following conditions holds:

1. 11DA1A704FF9

   $(\varphi _ i)_{i\in I}$ is decreasing and $\varphi $ is the limit of the net;

2. C18393916362

   $(\varphi _ i)_{i\in I}$ is increasing and converges almost everywhere to $\varphi $.

Then $(\varphi _ i)_{i\in I}$ converges to $\varphi $ in capacity.