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.
Contents
Definition
A subset
such that the
is zero. The backslash here denotes the set difference. Equivalently, the
A set
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
m rectifiable when there exists a Lipschitz mapf : K → E for some bounded subsetK ofR m E . - E is countably
m rectifiable when E equals the union of a countable family ofm rectifiable sets. - E is countably
( ϕ , m ) rectifiable whenϕ is a measure on X and there is a countablym rectifiable set F such thatϕ ( E ∖ F ) = 0 . - E is
( ϕ , m ) rectifiable when E is countably( ϕ , m ) rectifiable andϕ ( E ) < ∞ - E is purely
( ϕ , m ) unrectifiable whenϕ is a measure on X and E includes nom rectifiable set F withϕ ( F ) > 0 .
Definition 3 with