  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 the set of all bounded 1-Lipschitz functions
$f\colon X \to \mathbb R^m$ 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.
Conversely, if $E\subset X$ is $n$-rectifiable with $\mathcal H^n(E)>0$, the
set of all $f$ with $\mathcal H^n(f(E))>0$ is residual.
  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.
