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2001/2002 Sets Whose Hausdorff Measure Equals Method I Outer Measure
Richard Delaware
Real Anal. Exchange 27(2): 535-562 (2001/2002).


A set $E\subseteq \mathbb{R}^n $ is $s$-straight for $s>0$ if $E$ has finite Method II outer $s$-measure equal to its Method I outer $s$-measure. If $E$\ is Method II $s$-measurable this means $E$ has finite Hausdorff $s$-measure equal to its Hausdorff $s$-content. Here we make a first study of such sets, following their 1995 introduction by Foran. Primary facts are proved about subsets, intersections, unions, and some mappings of $s$-straight $s$-sets. Basic examples of $1$-straight and countable unions of $1$-straight $1$-sets are constructed from line segments. It is noted that self-similar $s $-sets are $s$-straight. Verifying a conjecture of Foran, the circle is proved to be the countable union of perfect $1$-straight $1$-sets along with a set of Hausdorff $1$-measure zero. Such perfect sets are then further examined. Also examined are subsets of $1$-straight sets $E$ maximal in the sense that their Hausdorff $1$-measure equals the diameter of $E$.


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Richard Delaware. "Sets Whose Hausdorff Measure Equals Method I Outer Measure." Real Anal. Exchange 27 (2) 535 - 562, 2001/2002.


Published: 2001/2002
First available in Project Euclid: 2 June 2008

zbMATH: 1044.28003
MathSciNet: MR1922667

Primary: 28A05 , 28A78

Keywords: $\sigma s$-straight , $s$-straight , Hausdorff

Rights: Copyright © 2001 Michigan State University Press

Vol.27 • No. 2 • 2001/2002
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