The Hilbert transform of a measure

Let e be a homogeneous subset of ℝ in the sense of Carleson. Let µ be a finite positive measure on ℝ and Hµ(x) its Hilbert transform. We prove that if limt→∞t|e∩{x ‖Hµ(x)| > t}| = 0, then µs(e) = 0, where µs is the singular part of µ.