A Borel subset ${\displaystyle E}$  of Euclidean space ${\displaystyle \mathbb {R} ^{n}}$  is said to be ${\displaystyle m}$ -rectifiable set if ${\displaystyle E}$  is of Hausdorff dimension ${\displaystyle m}$ , and there exist a countable collection ${\displaystyle \{f_{i}\}}$  of continuously differentiable maps

${\displaystyle f_{i}:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}$ 

such that the ${\displaystyle m}$ -Hausdorff measure ${\displaystyle {\mathcal {H}}^{m}}$  of

${\displaystyle E\setminus \bigcup _{i=0}^{\infty }f_{i}\left(\mathbb {R} ^{m}\right)}$ 

is zero. The backslash here denotes the set difference. Equivalently, the ${\displaystyle f_{i}}$  may be taken to be Lipschitz continuous without altering the definition.^[1][2][3] Other authors have different definitions, for example, not requiring ${\displaystyle E}$  to be ${\displaystyle m}$ -dimensional, but instead requiring that ${\displaystyle E}$  is a countable union of sets which are the image of a Lipschitz map from some bounded subset of ${\displaystyle \mathbb {R} ^{n}}$ .^[4]

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

${\displaystyle {\mathcal {H}}^{m}\left(E\cap f\left(\mathbb {R} ^{m}\right)\right)=0.}$ 

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 edit

Federer (1969, pp. 251–252) gives the following terminology for m-rectifiable sets E in a general metric space X.

1. E is ${\displaystyle m}$  rectifiable when there exists a Lipschitz map ${\displaystyle f:K\to E}$  for some bounded subset ${\displaystyle K}$  of ${\displaystyle \mathbb {R} ^{m}}$  onto ${\displaystyle E}$ .
2. E is countably ${\displaystyle m}$  rectifiable when E equals the union of a countable family of ${\displaystyle m}$  rectifiable sets.
3. E is countably ${\displaystyle (\phi ,m)}$  rectifiable when ${\displaystyle \phi }$  is a measure on X and there is a countably ${\displaystyle m}$  rectifiable set F such that ${\displaystyle \phi (E\setminus F)=0}$ .
4. E is ${\displaystyle (\phi ,m)}$  rectifiable when E is countably ${\displaystyle (\phi ,m)}$  rectifiable and ${\displaystyle \phi (E)<\infty }$ 
5. E is purely ${\displaystyle (\phi ,m)}$  unrectifiable when ${\displaystyle \phi }$  is a measure on X and E includes no ${\displaystyle m}$  rectifiable set F with ${\displaystyle \phi (F)>0}$ .

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

• Federer, Herbert (1969), Geometric measure theory, Die Grundlehren der mathematischen Wissenschaften, vol. 153, New York: Springer-Verlag, pp. xiv+676, ISBN 978-3-540-60656-7, MR 0257325
• T.C.O'Neil (2001) [1994], "Geometric measure theory", Encyclopedia of Mathematics, EMS Press
• Simon, Leon (1984), Lectures on Geometric Measure Theory, Proceedings of the Centre for Mathematical Analysis, vol. 3, Canberra: Centre for Mathematics and its Applications (CMA), Australian National University, pp. VII+272 (loose errata), ISBN 0-86784-429-9, Zbl 0546.49019

• Rectifiable set at Encyclopedia of Mathematics