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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
:<math>0 < \mu(\mathbf{B}_{r}(x)) = \mu(\mathbf{B}_{r}(y)) < + \infty</math>
for all points ''x'' and ''y'' of ''X'' and all 0&nbsp;&lt;&nbsp;''r''&nbsp;&lt;&nbsp;+&infin;, where
:<math>\mathbf{B}_{r}(x) := \{ z \in X | d(x, z) < r \}.</math>


==Christensen&rsquo;s lemma==


As it turns out, uniformly distributed measures are very rigid objects.  On any &ldquo;decent&rdquo; metric space, the uniformly distributed measures form a one-parameter linearly dependent family:
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Let ''&mu;'' and ''&nu;'' be uniformly distributed Borel regular measures on a [[separable space|separable]] metric space (''X'',&nbsp;''d'').  Then there is a constant ''c'' such that ''&mu;''&nbsp;=&nbsp;''c&nu;''.
 
==References==
 
* {{cite journal
| last = Christensen
| first = Jens Peter Reus
| title = On some measures analogous to Haar measure
| journal = Mathematica Scandinavica
| volume = 26
| year = 1970
| pages = 103&ndash;106
| issn = 0025-5521
}} {{MathSciNet|id=0260979}}
* {{cite book
| last = Mattila
| first = Pertti
| title = Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability
| series = Cambridge Studies in Advanced Mathematics No. 44
| publisher = Cambridge University Press
| location = Cambridge
| year = 1995
| pages = xii+343
| isbn = 0-521-46576-1
}} {{MathSciNet|id=1333890}} (See chapter 3)
 
[[Category:Measures (measure theory)]]

Latest revision as of 01:38, 7 October 2014


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