Lemma 8.2.10`A1FFEE061CF5`

Lemma 8.2.10.A1FFEE061CF5 If bounded admissible $u,v$ satisfy $u\leq v$ almost everywhere with respect to $\omega ^ n$, then $u\leq v$ everywhere.

Proof. By 8.2.9, $w=\max \{ u,v\} $ is bounded admissible with its indicated canonical representative. Since $w=v$ almost everywhere, they represent the same distribution. The canonical subharmonic representative used in (8.2.2) is determined by that distribution. Thus $w=v$ pointwise, which is the assertion. $\square$

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@misc{pome-tag-A1FFEE061CF5,
  author = {{Pome contributors}},
  title = {Lemma 8.2.10},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/A1FFEE061CF5/}
}

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Lemma 8.2.10A1FFEE061CF5

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Lemma 8.2.10`A1FFEE061CF5`