Let $\fre$ be a homogeneous subset of $\bbR$ in the sense of Carleson. Let $��$ be a finite positive measure on $\bbR$ and $H_��(x)$ its Hilbert transform. We prove that if $\lim_{t\to\infty} t \abs{\fre\cap\{x\mid\abs{H_��(x)}>t\}}=0$, then $��_s(\fre)=0$, where $��_\s$ is the singular part of $��$.

18 pages