Corollary 13.13.61`77CAD5891555`

Corollary 13.13.61.77CAD5891555 Assume that $\mathrm{PSH}(X,\theta )\neq \varnothing $. For any $f\in C^{1,\overline{1}}(X)$, we have

\[ \theta ^ n_{P_\theta (f)} = \mathds {1}_{\{ P_\theta (f)=f\} }\, \theta ^ n_ f. \]

In particular,

\[ \theta _{V_{\theta }}^ n=\mathds {1}_{\{ V_{\theta }=0\} }\, \theta ^ n. \]

Proof. From theorem 13.13.57, we find

\[ \mathds {1}_{\{ P_\theta (f)=f\} }\, \theta ^ n_{P_\theta (f)}=\mathds {1}_{\{ P_\theta (f)=f\} }\, \theta ^ n_ f. \]

By the concentration property theorem 13.13.24, our assertion follows. $\square$

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BibTeX citation

@misc{pome-tag-77CAD5891555,
  author = {{Pome contributors}},
  title = {Corollary 13.13.61},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/77CAD5891555/}
}

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Corollary 13.13.6177CAD5891555

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Corollary 13.13.61`77CAD5891555`