Let $E$ be a homogeneous subset of $\mathbb{R}$ in the
sense of Carleson. Let $\mu$ be a finite positive measure on
$\mathbb{R}$ and $H_\mu(x)$ its Hilbert transform. We prove that
$\lim_{t\to\infty} t|E\cap\{x : |H_\mu(x)|>t\}|=0$ if and only if
$\mu_s(E)=0$, where $\mu_s$ is the singular part of $\mu$.