Rectifiable set
From Wikipedia, the free encyclopedia
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.
[edit] Definition
A subset E of Euclidean space 
is said to be m-rectifiable set if there exist a collection {fi} of continuously differentiable maps
such that the m-Hausdorff measure 
of
is zero. The backslash here denotes the set difference. Equivalently, the fi may be taken to be Lipschitz continuous without altering the definition.
A set is said to be purely m-unrectifiable if for every (continuous, differentiable) 
, one has
.
A standard example of a purely-1-unrectifiable set in two dimensions is the cross-product of the Cantor set times itself.
[edit] References
- T.C.O'Neil (2001), “Geometric measure theory”, in Hazewinkel, Michiel, Encyclopaedia of Mathematics, Kluwer Academic Publishers, ISBN 978-1556080104
