# Purely unrectifiable metric spaces and perturbations of Lipschitz functions

## David Bate

created by dimarino on 18 Sep 2018

Università di Catania

Abstract.

We characterise purely $n$-unrectifiable subsets of a complete metric space with finite $n$-dimensional Hausdorff measure by studying non-linear projections (i.e. 1-Lipschitz functions) into some fixed Euclidean space. We will show that a typical (in the sense of Baire category) non-linear projection maps $S$ to a set of zero $n$-dimensional Hausdorff measure. Conversely, a typical non-linear projection maps an $n$-rectifiable subset to a set of positive $n$-dimensional Hausdorff measure.

These results provide a replacement for the classical Besicovitch-Federer projection theorem, which is known to be false outside of Euclidean spaces.

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