The authors study countably \(\mathcal H^k\)-rectifiable sets in metric spaces, that is, Borel sets which can be covered, up to sets with \(\mathcal H^k\)-measure zero, by a countable family of Lipschitz images of subsets of \(\mathbb R^k\), and prove that many properties of rectifiable subsets of Euclidean spaces remain true in the more general metric setting. Here \(\mathcal H^k\) is the \(k\)-dimensional Hausdorff measure.
Among the several interesting results proved in the paper under review are a description of the local metric behaviour of countably \(\mathcal H^k\)-rectifiable sets with finite \(\mathcal H^k\)-measure in terms of approximate tangent spaces, the area and coarea formulas for Lipschitz maps defined on countably \(\mathcal H^k\)-rectifiable subsets of a metric space, and rectifiability criteria for sets and measures in dual Banach spaces. The authors also give examples of purely and strongly \(k\)-unrectifiable metric spaces. A metric space \(E\) is purely \(k\)-unrectifiable if \(\mathcal H^k(S)=0\) for all countably \(\mathcal H^k\)-rectifiable subsets \(S\) of \(E\). Furthermore, \(E\) is strongly \(k\)-unrectifiable if \(\mathcal H^k(E)<\infty\) and \(\mathcal H^k(f(E))=0\) for all Lipschitz maps with values in a Euclidean space.