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Every set of finite Hausdorff measure is a countable union of sets whose Hausdorff measure and content coincide


Author: Richard Delaware
Journal: Proc. Amer. Math. Soc. 131 (2003), 2537-2542
MSC (2000): Primary 28A78, 28A05, 28A12
Published electronically: November 13, 2002
MathSciNet review: 1974652
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Abstract: A set $E\subseteq \mathbb{R} ^{n}$ is $h$-straight if $E$ has finite Hausdorff $h$-measure equal to its Hausdorff $h$-content, where $h:[0,\infty )\rightarrow \lbrack 0,\infty )$ is continuous and non-decreasing with $h(0)=0$. Here, if $h$ satisfies the standard doubling condition, then every set of finite Hausdorff $h$-measure in $\mathbb{R} ^{n}$ is shown to be a countable union of $h$-straight sets. This also settles a conjecture of Foran that when $h(t)=t^{s}$, every set of finite $s$-measure is a countable union of $s$-straight sets.


References [Enhancements On Off] (What's this?)

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Additional Information

Richard Delaware
Affiliation: Department of Mathematics and Statistics, Haag Hall Room 206, University of Missouri - Kansas City, 5100 Rockhill Rd., Kansas City, Missouri 64110
Email: RDelaware3141@cs.com

DOI: https://doi.org/10.1090/S0002-9939-02-06825-9
Keywords: $h$-straight, Hausdorff measure, Hausdorff content
Received by editor(s): August 17, 2001
Received by editor(s) in revised form: March 27, 2002
Published electronically: November 13, 2002
Communicated by: David Preiss
Article copyright: © Copyright 2002 American Mathematical Society