In mathematics, a rectifiable set is a set that is smooth in a certain measure-theoretic sense. It is an extension of the idea of a rectifiable curve to higher dimensions; loosely speaking, a rectifiable set is a rigorous formulation of a piece-wise smooth set. As such, it has many of the desirable properties of smooth manifolds, including tangent spaces that are defined almost everywhere. Rectifiable sets are the underlying object of study in geometric measure theory.
such that the -Hausdorff measure of
A set is said to be purely -unrectifiable if for every (continuous, differentiable) , one has
A standard example of a purely-1-unrectifiable set in two dimensions is the cross-product of the Smith–Volterra–Cantor set times itself.
Rectifiable sets in metric spaces
Federer (1969, pp. 251–252) gives the following terminology for m-rectifiable sets E in a general metric space X.
- E is rectifiable when there exists a Lipschitz map for some bounded subset of onto .
- E is countably rectifiable when E equals the union of a countable family of rectifiable sets.
- E is countably rectifiable when is a measure on X and there is a countably rectifiable set F such that .
- E is rectifiable when E is countably rectifiable and
- E is purely unrectifiable when is a measure on X and E includes no rectifiable set F with .
Definition 3 with and comes closest to the above definition for subsets of Euclidean spaces.
- Simon 1984, p. 58, calls this definition "countably m-rectifiable".
- Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften 153, New York: Springer-Verlag, pp. xiv+676, ISBN 978-3-540-60656-7, MR 0257325
- T.C.O'Neil (2001), "Geometric measure theory", in Hazewinkel, Michiel, Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4
- Simon, Leon (1984), Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis 3, Canberra: Centre for Mathematics and its Applications (CMA), Australian National University, pp. VII+272 (loose errata), ISBN 0-86784-429-9, Zbl 0546.49019