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

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.

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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

Information

Received: 16 January 2018; Revised: 26 March 2019; Published: March 2020
First available in Project Euclid: 16 January 2021

Digital Object Identifier: 10.4310/ACTA.2020.v224.n1.a1

Rights: Copyright © 2020 Institut Mittag-Leffler

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Vol.224 • No. 1 • March 2020
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