An elementary rectifiability lemma and some applications

TL;DR

This paper extends a classical theorem to show that certain sets in Euclidean and metric spaces contain unrectifiable subsets, providing tools to analyze the structure of singular sets in geometric analysis.

Contribution

It generalizes Besicovitch's theorem, establishing conditions under which sets contain unrectifiable subsets and characterizing rectifiability in metric spaces.

Findings

Sets not $\sigma$-finite in $\mathcal{H}^k$ contain unrectifiable subsets.

Souslin sets with all finite measure closed subsets being rectifiable are themselves rectifiable.

The theorem applies to a class of metric spaces, broadening its applicability.

Abstract

We generalize a classical theorem of Besicovitch, showing that, for any positive integers $k<n$, if $E\subset \mathbb R^n$ is a Souslin set which is not $\mathcal{H}^k$-$\sigma$-finite, then $E$ contains a purely unrectifiable closed set $F$ with $0< \mathcal{H}^k (F) < \infty$. Therefore, if $E\subset \mathbb R^n$ is a Souslin set with the property that every closed subset with finite $\mathcal{H}^k$ measure is $k$-rectifiable, then $E$ is $k$-rectifiable. We also point out that this theorem holds in a suitable class of metric spaces. Our interest is motivated by recent studies of the structure of the singular sets of several objects in geometric analysis and we explain the usefulness of our lemma with some examples.