Arkiv för Matematik

  • Ark. Mat.
  • Volume 48, Number 1 (2010), 149-176.

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

Claude Tricot

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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)=VolD(BV)diam(B)α-D and φα(B)=VolD(BV)α/D, where VolD denotes the D-dimensional volume.

Article information

Source
Ark. Mat., Volume 48, Number 1 (2010), 149-176.

Dates
Received: 21 January 2008
First available in Project Euclid: 31 January 2017

Permanent link to this document
https://projecteuclid.org/euclid.afm/1485907100

Digital Object Identifier
doi:10.1007/s11512-008-0087-8

Mathematical Reviews number (MathSciNet)
MR2594591

Zentralblatt MATH identifier
1196.28010

Rights
2008 © Institut Mittag-Leffler

Citation

Tricot, Claude. General Hausdorff functions, and the notion of one-sided measure and dimension. Ark. Mat. 48 (2010), no. 1, 149--176. doi:10.1007/s11512-008-0087-8. https://projecteuclid.org/euclid.afm/1485907100


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