  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.
