  The classical Besicovitch-Federer projection theorem implies that the
d-dimensional Hausdorff measure of a set in Euclidean space with non-negligible
d-unrectifiable part will strictly decrease under orthogonal projection onto
almost every d-dimensional linear subspace. In fact, there exist maps which are
arbitrarily close to the identity in the C^0 topology which have the same
property. A converse holds as well, yielding the following rectifiability
criterion: under mild assumptions, a set is rectifiable if and only if its
Hausdorff measure is lower semi-continuous under bounded Lipschitz
perturbations.
