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.