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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



On the behavior of meromorphic functions at the ideal boundary of a Riemann surface

Author: J. L. Schiff
Journal: Proc. Amer. Math. Soc. 54 (1976), 130-132
MathSciNet review: 0390209
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Abstract: In a former work the author established an analog of a classical theorem of Painlevé in the context of an arbitrary resolutive compactification of a Riemann surface. In the same setting, a refinement of the argument used in the above yields an elementary proof of a theorem of Riesz-Luzin-Privaloff type: If a meromorphic function $ f$ tends to zero at each point of a subset $ E$ of the ideal boundary and $ E$ has positive harmonic measure, then $ f \equiv 0$ on $ R$. The well-known inclusion relations $ {U_{HB}} \subset {\mathcal{O}_{M{B^{\ast}}}}$ and $ {U_{HD}} \sim {\mathcal{O}_{M{D^{\ast}}}}$, are then established from the point of view of the resolutivity of the Wiener and Royden compactification respectively.

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Keywords: Resolutive compactification, ideal boundary, meromorphic function, superharmonic function, harmonic measure
Article copyright: © Copyright 1976 American Mathematical Society