Uniformly distributed measure: Difference between revisions

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:<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:
 
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;''.