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

Abstract

For f f meromorphic on Δ \Delta , let f ∗ {f^ * } denote the radial limit function of f f , defined at each point of M R {\mathcal {M}_R} where the limit exists. Let M R {\mathcal {M}_R} denote the class of functions for which f ∗ {f^ * } exists in a residual subset of C C . We prove the following theorem closely related to the Lusin-Privalov radial uniqueness theorem and its converse. There exists a nonconstant function f f in M R {\mathcal {M}_R} such that f ∗ ( η ) = 0 {f^ * }\left ( \eta \right ) = 0 , η ∈ E \eta \in E , if and only if E E is not metrically dense in any open arc of C C . We then show that sufficiency can be proved using functions whose moduli have radial limits at each point of C C .