Restricted Hausdorff Content, Frostman’s Lemma and Choquet Integrals

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• 作者：Safari Mukeru
• 关键词：Hausdorff dimension ; Hausdorff content ; Frostman lemma ; Borel measure ; Capacity ; Choquet integral ; 28A78 ; 28A12 ; 31C15
• 刊名：Bulletin of the Malaysian Mathematical Sciences Society
• 出版年：2015
• 出版时间：July 2015
• 年：2015
• 卷：38
• 期：3
• 页码：885-895
• 全文大小：432 KB
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• 作者单位：Safari Mukeru (1)
  
  1. Department of Decision Sciences, University of South Africa, P. O. Box 392, Pretoria, 0003, South Africa
  
• 刊物类别：Mathematics, general; Applications of Mathematics;
• 刊物主题：Mathematics, general; Applications of Mathematics;
• 出版者：Springer Singapore
• ISSN：2180-4206

文摘

In this paper, we extend the well-known Frostman lemma by showing that for any subset \(E\) of \([0, 1]\) and \(\alpha >0\), if the \(\alpha \)-Hausdorff measure of \(E\) is positive, then there exist a non-zero Borel measure \(\mu \) on \([0, 1]\), a constant \(C>0\) and a subset \(E_0\) of \(E\) such that \(\mu (I) \le C \vert I \vert ^{\alpha }\) for any interval \(I\) and \(E_0\) is dense in the support of \(\mu \). Under an additional condition on \(E_0\), we show that \(\mu (B) = \mu [0, 1]\) for any Borel subset \(B\) containing \(E\). Using the notion of Choquet integral, we extend the notion of capacitarian dimension to arbitrary subset of \([0, 1]\) and prove a generalisation of Frostman’s theorem. Keywords Hausdorff dimension Hausdorff content Frostman lemma Borel measure Capacity Choquet integral