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

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.