Conditions for equality of Hausdorff and packing measures on ℝ

Abstract

This note answers the question, for which Hausdorff functions \(h\) may the \(h\)-Hausdorff and \(h\)-packing measures agree on some subset \(A\) of \(\mathbb{R}^n\), and be positive and finite. We show that these conditions imply that \(h\) is a regular density function, in the sense of Preiss, and that for each such function there is a subset of \(\mathbb{R}^n\) on which the \(h\)-Hausdorff and \(h\)-packing measures agree and are positive and finite.