Calculus of Variations and Geometric Measure Theory

C. De Lellis - I. Fleschler

An elementary rectifiability lemma and some applications

created by delellis on 06 Jul 2023



Inserted: 6 jul 2023
Last Updated: 6 jul 2023

Year: 2023


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. 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.

Keywords: Rectifiable sets, Hausdorff measure, Non-sigma finite sets, Quantitative rectifiability, Structure of singular sets