  Let $\varphi$ be a function in the complex Sobolev space $W^*(U)$, where $U$
is an open subset in $\mathbb{C}^k$. We show that the complement of the set of
Lebesgue points of $\varphi$ is pluripolar. The key ingredient in our approach
is to show that $|\varphi|^\alpha $ for $\alpha \in [1,2)$ is locally bounded
from above by a plurisubharmonic function.
