Corollary 15.37.24`D3170E7413EC`

Corollary 15.37.24.D3170E7413EC Under the hypotheses of proposition 15.37.21, if $u_ j\to u\in \mathrm{PSH}(X,\theta )$ in $L^1(X)$, then $u\in \mathcal E_\varphi $.

Proof. The common envelope gives $h\leq u_ j\leq \varphi +C$. Along an almost-everywhere convergent subsequence, lemma 5.24.3 gives $h\leq u\leq \varphi +C$, and upward stability gives the conclusion. $\square$

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@misc{pome-tag-D3170E7413EC,
  author = {{Pome contributors}},
  title = {Corollary 15.37.24},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/D3170E7413EC/}
}

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Corollary 15.37.24D3170E7413EC

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Corollary 15.37.24`D3170E7413EC`