- Rectifiable set
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 arectifiable 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 smoothmanifold s, including tangent spaces that are definedalmost everywhere . Rectifiable sets are the underlying object of study ingeometric measure theory .Definition
A subset of
Euclidean space is said to be -rectifiable set if there exist a collection of continuously differentiable maps:
such that the -
Hausdorff measure of:
is zero. The backslash here denotes the
set difference . Equivalently, the may be taken to beLipschitz continuous without altering the definition.A set is said to be purely -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.References
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