Proposition 5.14.6`FA3AF40B9A0A`

Proposition 5.14.6.FA3AF40B9A0A Suppose that $\varphi ,\psi \in \mathrm{PSH}(X)$. If $\varphi \leq \psi $ almost everywhere, then $\varphi \leq \psi $ everywhere.

In particular, if $\varphi =\psi $ almost everywhere, then $\varphi =\psi $ everywhere.

Proof. Replace $\varphi $ by $\varphi \lor \psi $. Then $\varphi ,\psi $ are two plurisubharmonic functions that coincide almost everywhere. The uniqueness statement in theorem 5.13.18 gives $\varphi =\psi $. $\square$

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BibTeX citation

@misc{pome-tag-FA3AF40B9A0A,
  author = {{Pome contributors}},
  title = {Proposition 5.14.6},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/FA3AF40B9A0A/}
}

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Proposition 5.14.6FA3AF40B9A0A

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Proposition 5.14.6`FA3AF40B9A0A`