Purely unrectifiable metric spaces and perturbations of Lipschitz functions

Abstract

We characterise purely $n$-unrectifiable subsets $S$ of a complete metric space $X$ with finite Hausdorff $n$-measure by studying arbitrarily small perturbations of elements of $\operatorname{Lip}_1 (X, m)$, the set of all bounded $1$-Lipschitz functions $f : X \to \mathbb{R}^m$ with respect to the supremum norm. In one such characterisation it is shown that, if $S$ has positive lower density almost everywhere, then the set of all $f$ with $\mathcal{H}^n (f(S)) = 0$ is residual in $\operatorname{Lip}_1 (X, m)$. Conversely, if $E \subset X$ is $n$-rectifiable with $\mathcal{H}^n (E) \gt 0$, the set of all $f$ with $\mathcal{H}^n (f(E)) \gt 0$ is residual in $\operatorname{Lip}_1 (X, m)$.

These results provide a replacement for the Besicovitch–Federer projection theorem in arbitrary metric spaces, which is known to be false outside of Euclidean spaces.