# Rectifiable set

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{{Short description|Mathematics concept}}
{{About|rectifiable sets in measure theory|rectifiable curves|Arc length}}
In [mathematics](/source/mathematics), a '''rectifiable set''' is a set that is smooth in a certain [measure-theoretic](/source/measure_theory) sense. It is an extension of the idea of a [rectifiable curve](/source/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 [manifold](/source/manifold)s, including tangent spaces that are defined [almost everywhere](/source/almost_everywhere). Rectifiable sets are the underlying object of study in [geometric measure theory](/source/geometric_measure_theory).

==Definition==
A [Borel subset](/source/Borel_set) <math>E</math> of [Euclidean space](/source/Euclidean_space) <math>\mathbb{R}^n</math> is said to be '''<math>m</math>-rectifiable''' set if <math>E</math> is of [Hausdorff dimension](/source/Hausdorff_dimension) <math>m</math>, and there exist a [countable](/source/countable) collection <math>\{f_i\}</math> of continuously differentiable maps

:<math>f_i:\mathbb{R}^m \to \mathbb{R}^n</math>

such that the <math>m</math>-[Hausdorff measure](/source/Hausdorff_measure) <math>\mathcal{H}^m</math> of

:<math>E\setminus \bigcup_{i=0}^\infty f_i\left(\mathbb{R}^m\right)</math>

is zero. The backslash here denotes the [set difference](/source/set_difference). Equivalently, the <math>f_i</math> may be taken to be [Lipschitz continuous](/source/Lipschitz_continuous) without altering the definition.<ref>{{harvnb|Simon|1984|p=58}}, calls this definition "countably ''m''-rectifiable".</ref><ref>{{SpringerEOM|title=Rectifiable set|id=Rectifiable_set&oldid=29261}}</ref><ref>{{MathWorld|title=Rectifiable Set|id=RectifiableSet|access-date=2020-04-17}}</ref> Other authors have different definitions, for example, not requiring <math>E</math> to be <math>m</math>-dimensional, but instead requiring that <math>E</math> is a  countable union of sets which are the image of a Lipschitz map from some bounded subset of <math>\mathbb{R}^m</math>.<ref>{{harvtxt|Federer|1969|pp=3.2.14}}</ref>

A set <math>E</math> is said to be '''purely <math>m</math>-unrectifiable''' if for ''every'' (continuous, differentiable)  <math>f:\mathbb{R}^m \to \mathbb{R}^n</math>, one has

:<math>\mathcal{H}^m \left(E \cap f\left(\mathbb{R}^m\right)\right)=0.</math>

A standard example of a purely-1-unrectifiable set in two dimensions is the Cartesian product of the [Smith–Volterra–Cantor set](/source/Smith%E2%80%93Volterra%E2%80%93Cantor_set) times itself.

=== Rectifiable sets in metric spaces ===

{{harvtxt|Federer|1969|pp=251–252}} gives the following terminology for ''m''-rectifiable sets ''E'' in a general metric space ''X''.
# ''E'' is '''<math>m</math> rectifiable''' when there exists a Lipschitz map <math>f:K \to E</math> for some bounded subset <math>K</math> of <math>\mathbb{R}^m</math> onto <math>E</math>.
# ''E'' is '''countably <math>m</math> rectifiable''' when ''E'' equals the union of a countable family of <math>m</math> rectifiable sets.
# ''E'' is '''countably <math>(\phi,m)</math> rectifiable''' when <math>\phi</math> is a measure on ''X'' and there is a countably <math>m</math> rectifiable set ''F'' such that <math>\phi(E\setminus F)=0</math>.
# ''E'' is '''<math>(\phi,m)</math> rectifiable''' when ''E'' is countably <math>(\phi,m)</math> rectifiable and <math>\phi(E)<\infty</math>
# ''E'' is '''purely <math>(\phi,m)</math> unrectifiable''' when <math>\phi</math> is a measure on ''X'' and ''E'' includes no <math>m</math> rectifiable set ''F'' with <math>\phi(F)>0</math>.

Definition 3 with <math>\phi=\mathcal{H}^m</math> and <math>X=\mathbb{R}^n</math> comes closest to the above definition for subsets of Euclidean spaces.

==Notes==
{{reflist}}

==References==
* {{citation|last = Federer | first = Herbert | authorlink = Herbert Federer | title = Geometric measure theory| publisher = Springer-Verlag | location = New York | year = 1969 | pages = xiv+676 | isbn = 978-3-540-60656-7 | mr= 0257325 | series = Die Grundlehren der mathematischen Wissenschaften|volume=153}}
* {{springer|author=T.C.O'Neil|id=G/g130040|title=Geometric measure theory}}
* {{Citation| last = Simon
  | first = Leon
  | author-link =Leon Simon
  | title = Lectures on Geometric Measure Theory
  | place = [Canberra](/source/Canberra)
  | publisher = Centre for Mathematics and its Applications (CMA), [Australian National University](/source/Australian_National_University)
  | series = Proceedings of the Centre for Mathematical Analysis
  | volume = 3
  | year = 1984
  | pages =VII+272 (loose errata)
  | isbn = 0-86784-429-9
  | zbl = 0546.49019
}}

==External links==
* [https://www.encyclopediaofmath.org/index.php/Rectifiable_set Rectifiable set] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]

Category:Measure theory

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Adapted from the Wikipedia article [Rectifiable set](https://en.wikipedia.org/wiki/Rectifiable_set) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Rectifiable_set?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.