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| In [[mathematics]] — specifically, in [[geometric measure theory]] — a '''uniformly distributed measure''' on a [[metric space]] is one for which the measure of an [[open ball]] depends only on its radius and not on its centre. By convention, the measure is also required to be [[Borel regular measure|Borel regular]], and to take positive and finite values on open balls of finite radius. Thus, if (''X'', ''d'') is a metric space, a Borel regular measure ''μ'' on ''X'' is said to be '''uniformly distributed''' if
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| :<math>0 < \mu(\mathbf{B}_{r}(x)) = \mu(\mathbf{B}_{r}(y)) < + \infty</math>
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| for all points ''x'' and ''y'' of ''X'' and all 0 < ''r'' < +∞, where
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| :<math>\mathbf{B}_{r}(x) := \{ z \in X | d(x, z) < r \}.</math>
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| ==Christensen’s lemma==
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| As it turns out, uniformly distributed measures are very rigid objects. On any “decent” metric space, the uniformly distributed measures form a one-parameter linearly dependent family:
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| Let ''μ'' and ''ν'' be uniformly distributed Borel regular measures on a [[separable space|separable]] metric space (''X'', ''d''). Then there is a constant ''c'' such that ''μ'' = ''cν''.
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| ==References==
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| * {{cite journal
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| | last = Christensen
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| | first = Jens Peter Reus
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| | title = On some measures analogous to Haar measure
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| | journal = Mathematica Scandinavica
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| | volume = 26
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| | year = 1970
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| | pages = 103–106
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| | issn = 0025-5521
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| }} {{MathSciNet|id=0260979}}
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| * {{cite book
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| | last = Mattila
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| | first = Pertti
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| | title = Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability
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| | series = Cambridge Studies in Advanced Mathematics No. 44
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| | publisher = Cambridge University Press
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| | location = Cambridge
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| | year = 1995
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| | pages = xii+343
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| | isbn = 0-521-46576-1
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| }} {{MathSciNet|id=1333890}} (See chapter 3)
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| [[Category:Measures (measure theory)]]
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