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A note on the Lusin-Privalov radial uniqueness theorem and its converse

Author: Robert D. Berman
Journal: Proc. Amer. Math. Soc. 92 (1984), 64-66
MSC: Primary 30D40
MathSciNet review: 749892
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Abstract: For $ f$ meromorphic on $ \Delta $, let $ {f^ * }$ denote the radial limit function of $ f$, defined at each point of $ {\mathcal{M}_R}$ where the limit exists. Let $ {\mathcal{M}_R}$ denote the class of functions for which $ {f^ * }$ exists in a residual subset of $ C$. We prove the following theorem closely related to the Lusin-Privalov radial uniqueness theorem and its converse. There exists a nonconstant function $ f$ in $ {\mathcal{M}_R}$ such that $ {f^ * }\left( \eta \right) = 0$, $ \eta \in E$, if and only if $ E$ is not metrically dense in any open arc of $ C$. We then show that sufficiency can be proved using functions whose moduli have radial limits at each point of $ C$.

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Keywords: Radial uniqueness, residual set, metric density
Article copyright: © Copyright 1984 American Mathematical Society

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