Purely unrectifiable metric spaces and perturbations of Lipschitz functions

TL;DR

This paper characterizes purely unrectifiable sets in metric spaces by examining small perturbations of Lipschitz functions, providing a metric space analogue to the Besicovitch-Federer projection theorem.

Contribution

It offers new characterizations of unrectifiable sets via Lipschitz perturbations, extending geometric measure theory concepts beyond Euclidean spaces.

Findings

Residual sets of functions map unrectifiable sets to measure zero images.

Residual sets of functions map rectifiable sets to positive measure images.

Provides a metric space version of the Besicovitch-Federer projection theorem.

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 the set of all bounded 1-Lipschitz functions $f\colon X \to \mathbb R^m$ 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. Conversely, if $E\subset X$ is $n$-rectifiable with $\mathcal H^n(E)>0$, the set of all $f$ with $\mathcal H^n(f(E))>0$ is residual. 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.