Sets Whose Hausdorff Measure Equals Method I Outer Measure

Abstract

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 $E$.