Uniformly distributed measure
 Uniformly distributed measure

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, 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
for all points x and y of X and all 0 < r < +∞, where
Christensen’s lemma
As it turns out, uniformly distributed measures are very rigid objects. On any “decent” metric space, the uniformly distributed measures form a oneparameter linearly dependent family:
Let μ and ν be uniformly distributed Borel regular measures on a separable metric space (X, d). Then there is a constant c such that μ = cν.
References
 Christensen, Jens Peter Reus (1970). "On some measures analogous to Haar measure". Mathematica Scandinavica 26: 103–106. ISSN 00255521. MR0260979
 Mattila, Pertti (1995). Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability. Cambridge Studies in Advanced Mathematics No. 44. Cambridge: Cambridge University Press. pp. xii+343. ISBN 0521465761. MR1333890 (See chapter 3)
Categories:
 Measures (measure theory)
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