Lemma 15.24.4`F9088D3DDEBB`

Lemma 15.24.4.F9088D3DDEBB Let $\theta \geq 0$ be a smooth big real $(1,1)$-form, and let $\mu $ be a finite non-pluripolar measure on $X$. There are a bounded $\psi \in \mathrm{PSH}(X,\theta )$ and a nonnegative $f\in L^1(\mathrm{MA}_\theta (\psi ))$ such that

\[ \mu =f\, \mathrm{MA}_\theta (\psi ). \]

Proof. Apply proposition 15.24.1 with $\varphi =0$. Semipositivity of $\theta $ gives $P_{\epsilon \theta }(0)=0$ for every $0<\epsilon <1$, and $\psi \sim 0$ means that $\psi $ is bounded. $\square$

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@misc{pome-tag-F9088D3DDEBB,
  author = {{Pome contributors}},
  title = {Lemma 15.24.4},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/F9088D3DDEBB/}
}

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Lemma 15.24.4F9088D3DDEBB

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Lemma 15.24.4`F9088D3DDEBB`