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.
Citation
David Bate. "Purely unrectifiable metric spaces and perturbations of Lipschitz functions." Acta Math. 224 (1) 1 - 65, March 2020. https://doi.org/10.4310/ACTA.2020.v224.n1.a1
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