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
.