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Rectifiable sets in metric and Banach spaces. (English) Zbl 0966.28002
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.

##### MSC:
 28A75 Length, area, volume, other geometric measure theory 28A78 Hausdorff and packing measures 28A80 Fractals
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