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Restricted Hausdorff Content, Frostman’s Lemma and Choquet Integrals

Restricted Hausdorff Content, Frostman’s Lemma and Choquet Integrals In this paper, we extend the well-known Frostman lemma by showing that for any subset $$E$$ E of $$[0, 1]$$ [ 0 , 1 ] and $$\alpha >0$$ α > 0 , if the $$\alpha $$ α -Hausdorff measure of $$E$$ E is positive, then there exist a non-zero Borel measure $$\mu $$ μ on $$[0, 1]$$ [ 0 , 1 ] , a constant $$C>0$$ C > 0 and a subset $$E_0$$ E 0 of $$E$$ E such that $$\mu (I) \le C \vert I \vert ^{\alpha }$$ μ ( I ) ≤ C | I | α for any interval $$I$$ I and $$E_0$$ E 0 is dense in the support of $$\mu $$ μ . Under an additional condition on $$E_0$$ E 0 , we show that $$\mu (B) = \mu [0, 1]$$ μ ( B ) = μ [ 0 , 1 ] for any Borel subset $$B$$ B containing $$E$$ E . Using the notion of Choquet integral, we extend the notion of capacitarian dimension to arbitrary subset of $$[0, 1]$$ [ 0 , 1 ] and prove a generalisation of Frostman’s theorem. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Bulletin of the Malaysian Mathematical Sciences Society Springer Journals

Restricted Hausdorff Content, Frostman’s Lemma and Choquet Integrals

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References (21)

Publisher
Springer Journals
Copyright
Copyright © 2014 by Malaysian Mathematical Sciences Society and Universiti Sains Malaysia
Subject
Mathematics; Mathematics, general; Applications of Mathematics
ISSN
0126-6705
eISSN
2180-4206
DOI
10.1007/s40840-014-0055-3
Publisher site
See Article on Publisher Site

Abstract

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

Journal

Bulletin of the Malaysian Mathematical Sciences Society – Springer Journals

Published: Dec 4, 2014

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