The size of the set of μ-irregular points of a measure μ

Let μ be a compactly suppported positive measure on the real line. A point x∈supp [μ] is said to be μ-regular, if, as n → ∞, Otherwise it is a μ-irregular point. We show that for any such measure, the set of μ-irregular points in {μ′>0} (with a suitable definition of this set) has Hausdorff m_hβ measure 0, for h_β(t) = (log 1/t)^-β, any β>1.