  In this paper we prove that if $I $ is a subset of measure 0 in a
$C^2-$smooth generic submanifold $M$ of $ \C^n$, then its complement in $M$ is
non-plurithin at each point of $M$ in $\C^n$. This result improves a previous
result of A. Edigarian and J. Wiegerinck, who considered the case where $I$ is
pluripolar set contained in a $C^1-$smooth generic submanifold $M \subset
\C^n$. The proof of our result is essentially different.
