Rectifiable set


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.

Definition

A Borel subset of Euclidean space is said to be -rectifiable set if is of Hausdorff dimension, and there exist a countable collection of continuously differentiable maps
such that the -Hausdorff measure of
is zero. The backslash here denotes the set difference. Equivalently, the may be taken to be Lipschitz continuous without altering the definition. Other authors have different definitions, for example, not requiring to be -dimensional, but instead requiring that is a countable union of sets which are the image of a Lipschitz map from some bounded subset of.
A set is said to be purely -unrectifiable if for every , 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

gives the following terminology for m-rectifiable sets E in a general metric space X.
  1. E is rectifiable when there exists a Lipschitz map for some bounded subset of onto.
  2. E is countably rectifiable when E equals the union of a countable family of rectifiable sets.
  3. E is countably rectifiable when is a measure on X and there is a countably rectifiable set F such that.
  4. E is rectifiable when E is countably rectifiable and
  5. 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.