Corollary 15.26.3`AB5B44EC4BDD`

Corollary 15.26.3.AB5B44EC4BDD Let $\nu $ be a finite nonzero positive measure with $M_\rho (\nu )<\infty $. There are $c>0$ and a Hölder continuous $w\in \mathrm{PSH}(X,h)$ satisfying

\[ \sup _ Xw=0,\qquad \mathrm{MA}_ h(w)=c\nu , \qquad 0<\mathrm{MA}_ h(w)(X)<\infty . \]

Proof. By proposition 15.26.2, $\nu \leq (M_\rho (\nu )/k)\mathrm{MA}_ h(u_\rho )$, where $u_\rho $ is Hölder continuous with exponent $1/2$. Apply theorem 15.25.1. The equation and $0<c<\infty $ give its positive finite mass. $\square$

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@misc{pome-tag-AB5B44EC4BDD,
  author = {{Pome contributors}},
  title = {Corollary 15.26.3},
  howpublished = {Pome},
  url = {https://mingchenxia.github.io/Pome/tag/AB5B44EC4BDD/}
}

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Corollary 15.26.3AB5B44EC4BDD

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Corollary 15.26.3`AB5B44EC4BDD`