Lebesgue points of functions in the complex Sobolev space

Abstract

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