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

Claude Tricot Département de Mathématiques, Université Clermont-Ferrand II

TBD mathscidoc:1701.333161

Arkiv for Matematik, 48, (1), 149-176, 2008.1
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 φ_{α}, 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 ℝ^{$D$}, 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 φ_{α}. Two determining functions are considered, φ_{α}($B$)=Vol_{$D$}($B$∩$V$)diam($B$)^{α-$D$}and φ_{α}($B$)=Vol_{$D$}($B$∩$V$)^{α/$D$}, where Vol_{$D$}denotes the$D$-dimensional volume.
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@inproceedings{claude2008general,
  title={General Hausdorff functions, and the notion of one-sided measure and dimension},
  author={Claude Tricot},
  url={http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203626327628970},
  booktitle={Arkiv for Matematik},
  volume={48},
  number={1},
  pages={149-176},
  year={2008},
}
Claude Tricot. General Hausdorff functions, and the notion of one-sided measure and dimension. 2008. Vol. 48. In Arkiv for Matematik. pp.149-176. http://archive.ymsc.tsinghua.edu.cn/pacm_paperurl/20170108203626327628970.
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