Published Paper
Inserted: 6 jul 2023
Last Updated: 12 apr 2026
Journal: Math. Res. Lett.
Volume: 31
Number: 4
Pages: 1029-1046
Year: 2024
The second version contains minor corrections, an additional general statement in a class of metric spaces, and an outline of the argument for its validity
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. 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
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