rectifiable set

Let us denote with m the m-dimensional Hausdorff measureMathworldPlanetmath in n.

A set Sn is said to be countably m-rectifiable if there exists a countableMathworldPlanetmath sequence of Lipschitz continuous functions fk:mn such that


A set Sn is said to be countably (Hm,m)-rectifiable if there exists a set S which is countable m-rectifiable and such that m(SS)=0.

A set Sn is said to be (Hm,m)-rectifiable or simply m-rectifiable if it is (m,m)-rectifiable and m(S)<+.

If S is any Borel subset of n and xn is given, one can define the density of S in x as


where ωm is the Lebesgue measureMathworldPlanetmath of the unit ballMathworldPlanetmath in m. Notice that an m-dimensional plane Π has density 1 in all points xΠ and density 0 in all points xΠ.

It turns out that if S is rectifiable, then in m-a.e. point xS the density Θm(S,x) exists and is equal to 1. Moreover in m-a.e. point xS there exists an approximate tangent plane to S as defined below.

Given a point xSn and a vector vn we say that v is tangentPlanetmathPlanetmathPlanetmath to S in x if there exists a sequence of points xkS, xkx and a sequence of positive numbers λk such that


If S is an m-dimensional manifold, then the set of tangent vectors to a point xS is nothing else than the usual tangent planeMathworldPlanetmath to S in x.

We say that a vector v is approximately tangent to S in x if it is a tangent vector to every subset S of S such that Θm(SS,x)=0. Notice that every tangent vector is also an approximately tangent vector while the converseMathworldPlanetmath is not always true, as it is shown in an example below. The point, here, is that being the set S defined m-almost everywhere, we need a stronger definition for tangent vectors.

The approximate tangent cone to S in x is the set of all approximately tangent vectors to S in x (notice that if v is a tangent vector then λv is also a tangent vector, for all λ>0). If the approximate tangent cone is an m-dimensional vector subspace of n, it is called the approximate tangent plane.

Notice that if Sn is any m-dimensional regular surface, and Q is the set of all points of n with rational coordinates, then the set SQ is an m-rectifiable set since 1(Q)=0. Notice, however, that SQ¯=n and consequently every vector v is tangent to SQ in every point xSQ. On the other hand the approximately tangent vectors to SQ are only the tangent vectors to S, because the set Q has density 0 everywhere.


  • 1 Frank Morgan: Geometric Measure Theory: A Beginner’s Guide.
Title rectifiable set
Canonical name RectifiableSet
Date of creation 2013-03-22 14:28:12
Last modified on 2013-03-22 14:28:12
Owner paolini (1187)
Last modified by paolini (1187)
Numerical id 11
Author paolini (1187)
Entry type Definition
Classification msc 49Q15
Defines density
Defines tangent vector
Defines approximate tangent plane