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