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 E of Euclidean space is said to be m-rectifiable set if E is of Hausdorff dimension m, and there exist a countable collection {f_i} of continuously differentiable maps such that the m-Hausdorff measure of is zero. The backslash here denotes the set difference. Equivalently, the f_i may be taken to be Lipschitz continuous without altering the definition. Other authors have different definitions, for example, not requiring E to be m-dimensional, but instead requiring that E is a countable union of sets which are the image of a Lipschitz map from some bounded subset of. A set E is said to be purely m-unrectifiable if for every (continuous, differentiable), one has A standard example of a purely-1-unrectifiable set in two dimensions is the Cartesian 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. Definition 3 with and comes closest to the above definition for subsets of Euclidean spaces.