An elementary rectifiability lemma and some applications

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 $0<{\mathcal{H}}^k(F)<\mathrm{\infty }$. 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.