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| {{otheruses4|rectifiable sets in measure theory|rectifiable curves|Arc length}}
| | I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Mississippi is exactly where his home is. Invoicing is what I do. What me and my family members adore is bungee jumping but I've been taking on new things recently.<br><br>Feel free to visit my web page spirit messages; [http://1.234.36.240/fxac/m001_2/7330 http://1.234.36.240/fxac/m001_2/7330], |
| In [[mathematics]], a '''rectifiable set''' is a set that is smooth in a certain [[measure theory|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 [[manifold]]s, including tangent spaces that are defined [[almost everywhere]]. Rectifiable sets are the underlying object of study in [[geometric measure theory]].
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| ==Definition==
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| A subset <math>E</math> of [[Euclidean space]] <math>\mathbb{R}^n</math> is said to be '''<math>m</math>-rectifiable''' set if there exist a [[countable]] collection <math>\{f_i\}</math> of continuously differentiable maps
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| :<math>f_i:\mathbb{R}^m \to \mathbb{R}^n</math>
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| such that the <math>m</math>-[[Hausdorff measure]] <math>\mathcal{H}^m</math> of
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| :<math>E\backslash \bigcup_{i=0}^\infty f_i\left(\mathbb{R}^m\right)</math>
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| is zero. The backslash here denotes the [[set difference]]. Equivalently, the <math>f_i</math> may be taken to be [[Lipschitz continuous]] without altering the definition.
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| A set <math>E</math> is said to be '''purely <math>m</math>-unrectifiable''' if for ''every'' (continuous, differentiable) <math>f:\mathbb{R}^m \to \mathbb{R}^n</math>, one has
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| :<math>\mathcal{H}^m \left(E \cap f\left(\mathbb{R}^m\right)\right)=0.</math>
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| A standard example of a purely-1-unrectifiable set in two dimensions is the cross-product of the [[Smith-Volterra-Cantor set]] times itself.
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| ==References==
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| * {{springer|author=T.C.O'Neil|id=G/g130040|title=Geometric measure theory}}
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| ==External links==
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| * [http://www.encyclopediaofmath.org/index.php/Rectifiable_set Rectifiable set] at [http://www.encyclopediaofmath.org/ Encyclopedia of Mathematics]
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| [[Category:Measure theory]]
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Latest revision as of 07:20, 4 August 2014
I would like to introduce myself to you, I am Andrew and my wife doesn't like it at all. Mississippi is exactly where his home is. Invoicing is what I do. What me and my family members adore is bungee jumping but I've been taking on new things recently.
Feel free to visit my web page spirit messages; http://1.234.36.240/fxac/m001_2/7330,