- 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 E of
Euclidean space mathbb{R}^n is said to be m-rectifiable set if there exist a collection f_i} of continuously differentiable maps:f_i:mathbb{R}^m o mathbb{R}^n
such that the m-
Hausdorff measure mathcal{H}^m of:Eackslash igcup_{i=0}^infty f_ileft(mathbb{R}^m ight)
is zero. The backslash here denotes the
set difference . Equivalently, the f_i may be taken to beLipschitz continuous without altering the definition.A set is said to be purely m-unrectifiable if for "every" (continuous, differentiable) f:mathbb{R}^m o mathbb{R}^n, one has
:mathcal{H}^m left(E cap fleft(mathbb{R}^m ight) ight)=0.
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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