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
of Euclidean space
is said to be
-rectifiable set if
is of Hausdorff dimension
, and there exist a countable collection
of continuously differentiable maps
![{\displaystyle f_{i}:\mathbb {R} ^{m}\to \mathbb {R} ^{n}}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
such that the
-Hausdorff measure
of
![{\displaystyle E\setminus \bigcup _{i=0}^{\infty }f_{i}\left(\mathbb {R} ^{m}\right)}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
is zero. The backslash here denotes the set difference. Equivalently, the
may be taken to be Lipschitz continuous without altering the definition.[1][2][3] Other authors have different definitions, for example, not requiring
to be
-dimensional, but instead requiring that
is a countable union of sets which are the image of a Lipschitz map from some bounded subset of
.[4]
A set
is said to be purely
-unrectifiable if for every (continuous, differentiable)
, one has
![{\displaystyle {\mathcal {H}}^{m}\left(E\cap f\left(\mathbb {R} ^{m}\right)\right)=0.}](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
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
Federer (1969, pp. 251–252) gives the following terminology for m-rectifiable sets E in a general metric space X.
- E is
rectifiable when there exists a Lipschitz map
for some bounded subset
of
onto
. - E is countably
rectifiable when E equals the union of a countable family of
rectifiable sets. - E is countably
rectifiable when
is a measure on X and there is a countably
rectifiable set F such that
. - E is
rectifiable when E is countably
rectifiable and ![{\displaystyle \phi (E)<\infty }](data:image/gif;base64,R0lGODlhAQABAIAAAAAAAP///yH5BAEAAAAALAAAAAABAAEAAAIBRAA7)
- E is purely
unrectifiable when
is a measure on X and E includes no
rectifiable set F with
.
Definition 3 with
and
comes closest to the above definition for subsets of Euclidean spaces.
Notes
References
- 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
External links
- Rectifiable set at Encyclopedia of Mathematics