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General Hausdorff functions, and the notion of one-sided measure and dimension

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Arkiv för Matematik

Abstract

The main facts about Hausdorff and packing measures and dimensions of a Borel set E are revisited, using determining set functions \(\phi_\alpha\colon\mathcal{B}_E\to(0,\infty)\), where \(\mathcal{B}_E\) is the family of all balls centred on E and α is a real parameter. With mild assumptions on φα, we verify that the main density results hold, as well as the basic properties of the corresponding box dimension. Given a bounded open set V in ℝD, these notions are used to introduce the interior and exterior measures and dimensions of any Borel subset of ∂V. We stress that these dimensions depend on the choice of φα. Two determining functions are considered, φα(B)=Vol D (B∩V)diam(B)α-D and φα(B)=Vol D (B∩V)α/D, where Vol D denotes the D-dimensional volume.

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Authors and Affiliations

  1. Département de Mathématiques, Université Clermont-Ferrand II, FR-63177, Aubière, France

    Claude Tricot

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  1. Claude Tricot
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Correspondence to Claude Tricot.

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Tricot, C. General Hausdorff functions, and the notion of one-sided measure and dimension. Ark Mat 48, 149–176 (2010). https://doi.org/10.1007/s11512-008-0087-8

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  • DOI: https://doi.org/10.1007/s11512-008-0087-8

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