A Note on the Lusin-Privalov Radial Uniqueness Theorem and its Converse

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 $C$.