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| {{Orphan|date=October 2013}}
| | Friends call him Royal. Some time in the past he chose to live in Idaho. The favorite pastime for my kids and me is taking part in crochet and now I'm attempting to make cash with it. She works as a monetary officer and she will not alter it whenever soon.<br><br>Check out my blog post ... [http://Newdayz.de/index.php?mod=users&action=view&id=16038 http://Newdayz.de/index.php?mod=users&action=view&id=16038] |
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| In [[measure theory]], '''tangent measures''' are used to study the local behavior of [[Radon measure]]s, in much the same way as [[tangent space]]s are used to study the local behavior of [[differentiable manifold]]s. Tangent measures (introduced by David Preiss <ref>{{cite journal
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| | last = Preiss | first = David
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| | title = Geometry of measures in <math>R^n</math>: distribution, rectifiability, and densities
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| | journal = Ann. Math. | volume = 125
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| | issue = 3 | pages = 537–643 | year = 1987
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| | jstor = 1971410
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| | doi = 10.2307/1971410
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| }}</ref> in his study of [[rectifiable set]]s) are a useful tool in geometric measure theory. For example, they are used in proving [[Hausdorff density|Marstrand’s theorem]] and [[Hausdorff density|Preiss' theorem]].
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| ==Definition==
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| Consider a Radon measure ''μ'' defined on an [[open subset]] Ω of ''n''-dimensional [[Euclidean space]] '''R'''<sup>''n''</sup> and let ''a'' be an arbitrary point in Ω. We can “zoom in” on a small [[open ball]] of radius ''r'' around ''a'', ''B''<sub>''r''</sub>(''a''), via the transformation
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| : <math>T_{a,r}(x)=\frac{x-a}{r},</math>
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| which enlarges the ball of radius ''r'' about ''a'' to a ball of radius 1 centered at 0. With this, we may now zoom in on how ''μ'' behaves on ''B''<sub>''r''</sub>(''a'') by looking at the [[push-forward measure]] defined by
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| :<math> T_{a,r \#}\mu(A)=\mu(a+rA)</math>
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| where
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| :<math>a+rA=\{a+rx:x\in A\}.</math>
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| As ''r'' gets smaller, this transformation on the measure ''μ'' spreads out and enlarges the portion of ''μ'' supported around the point ''a''. We can get information about our measure around ''a'' by looking at what these measures tend to look like in the limit as ''r'' approaches zero.
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| :'''Definition.''' A ''tangent measure'' of a Radon measure ''μ'' at the point ''a'' is a second Radon measure ''ν'' such that there exist sequences of positive numbers ''c''<sub>''i''</sub> > 0 and decreasing radii ''r''<sub>''i''</sub> → 0 such that
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| :: <math>\lim_{i\rightarrow\infty} c_{i}T_{a,r_{i}\#}\mu =\nu </math>
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| : where the limit is taken in the [[weak-star topology|weak-∗ topology]], i.e., for any [[continuous function]] ''φ'' with [[compact support]] in Ω,
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| :: <math> \lim_{i\rightarrow\infty}\int_{\Omega} \varphi \, \mathrm{d} (c_{i}T_{a,r_{i}\#}\mu)=\int_{\Omega} \varphi \, \mathrm{d} \nu.</math>
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| :We denote the set of tangent measures of ''μ'' at ''a'' by Tan(''μ'', ''a'').
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| ==Existence==
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| The set Tan(''μ'', ''a'') of tangent measures of a measure ''μ'' at a point ''a'' in the [[support (measure theory)|support]] of ''μ'' is nonempty on mild conditions on ''μ''. By the weak compactness of Radon measures, Tan(''μ'', ''a'') is nonempty if one of the following conditions hold:
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| * ''μ'' is [[Doubling measures and metric spaces|asymptotically doubling]] at ''a'', i.e. <math>\limsup_{r\downarrow 0} \frac{\mu(B(a,2r))}{\mu(B(a,r))}<\infty</math>
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| * ''μ'' has positive and finite [[Hausdorff density|upper density]], i.e. <math>0<\limsup_{r\downarrow 0}\frac{\mu(B(a,r))}{r^s}<\infty</math> for some <math>0<s<\infty</math>.
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| ==Properties==
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| The collection of tangent measures at a point is closed under two types of scaling. Cones of measures were also defined by Preiss.
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| * The set Tan(''μ'', ''a'') of tangent measures of a measure ''μ'' at a point ''a'' in the support of ''μ'' is a ''cone'' of measures, i.e. if <math>\nu\in \mathrm{Tan}(\mu,a)</math> and <math>c>0</math>, then <math>c\nu\in \mathrm{Tan}(\mu,a)</math>.
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| * The cone Tan(''μ'', ''a'') of tangent measures of a measure ''μ'' at a point ''a'' in the support of ''μ'' is a ''d-cone'' or ''dilation invariant'', i.e. if <math>\nu\in \mathrm{Tan}(\mu,a)</math> and <math>r>0</math>, then <math>T_{0,r\#}\nu \in \mathrm{Tan}(\mu,a)</math>.
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| At typical points in the support of a measure, the cone of tangent measures is also closed under translations.
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| * At ''μ'' almost every ''a'' in the support of ''μ'', the cone Tan(''μ'', ''a'') of tangent measures of ''μ'' at ''a'' is ''translation invariant'', i.e. if <math>\nu\in\mathrm{Tan}(\mu,a)</math> and ''x'' is in the support of ''ν'', then <math>T_{x,1\#}\nu\in\mathrm{Tan}(\mu,a)</math>.
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| ==Examples==
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| *Suppose we have a circle in '''R'''<sup>2</sup> with uniform measure on that circle. Then, for any point ''a'' in the circle, the set of tangent measures will just be positive constants times 1-dimensional [[Hausdorff measure]] supported on the line tangent to the circle at that point.
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| * In 1995, Toby O'Neil produced an example of a Radon measure ''μ'' on '''R'''<sup>''d''</sup> such that, for μ-almost every point ''a'' ∈ '''R'''<sup>''d''</sup>, Tan(''μ'', ''a'') consists of all nonzero Radon measures.<ref>{{cite journal
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| | last = O'Neil | first = Toby
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| | title = A measure with a large set of tangent measures
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| | journal = Proc. of the AMS | volume = 123
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| | issue = 7 | pages = 2217–2220 | year = 1995
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| | doi = 10.2307/2160960
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| | jstor = 2160960
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| }}</ref>
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| ==Related concepts==
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| There is an associated notion of the tangent space of a measure. A ''k''-dimensional subspace ''P'' of '''R'''<sup>''n''</sup> is called the ''k''-dimensional tangent space of ''μ'' at ''a'' ∈ Ω if — after appropriate rescaling — ''μ'' “looks like” ''k''-dimensional [[Hausdorff measure]] ''H''<sup>''k''</sup> on ''P''. More precisely:
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| :'''Definition.''' ''P'' is the ''k''-''dimensional tangent space'' of ''μ'' at ''a'' if there is a ''θ'' > 0 such that
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| ::<math>\mu_{a, r} \xrightarrow[r \to 0]{*} \theta H^{k} \lfloor_{P},</math>
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| :where ''μ''<sub>''a'',''r''</sub> is the translated and rescaled measure given by
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| ::<math>\mu_{a, r} (A) = \frac1{r^{n - 1}} \mu(a + r A).</math>
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| :The number ''θ'' is called the ''multiplicity'' of ''μ'' at ''a'', and the tangent space of ''μ'' at ''a'' is denoted T<sub>''a''</sub>(''μ'').
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| Further study of tangent measures and tangent spaces leads to the notion of a [[varifold]].<ref>{{cite journal | last = Röger | first = Matthias | title = Solutions for the Stefan problem with Gibbs-Thomson law by a local minimisation | journal = Interfaces Free Bound. | volume = 6 | year = 2004 | issue = 1 | pages = 105–133 | issn = 1463-9963 }} {{MathSciNet|id=2047075}}</ref>
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| ==References==
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| <references />
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| [[Category:Measures (measure theory)]]
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Friends call him Royal. Some time in the past he chose to live in Idaho. The favorite pastime for my kids and me is taking part in crochet and now I'm attempting to make cash with it. She works as a monetary officer and she will not alter it whenever soon.
Check out my blog post ... http://Newdayz.de/index.php?mod=users&action=view&id=16038