Measure preserving continuous straightenings of fractional dimensional sets

Abstract

An \(s\)-set in Euclidean space is a set of finite, non-zero, Hausdorff \(s\)-dimensional measure. Call an \(s\)-set straight if its \(s\)-measure agrees with its Method I \(s\)-outer measure. Examples are given where there is a continuous, one-to-one function \(f\) on \(\mathbb{R}^n\) which is measure preserving on \( E\) so that \(f(E)\) is straight (such an \(f\) will be called a straightening of \(E\)). It is shown that any \(s\)-set can be written as a countable union of sets for which there are straightenings.