In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
— specifically, in
geometric measure theory
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfa ...
— a uniformly distributed measure on a
metric space
In mathematics, a metric space is a set together with a notion of ''distance'' between its elements, usually called points. The distance is measured by a function called a metric or distance function. Metric spaces are the most general settin ...
is one for which the measure of an
open ball
In mathematics, a ball is the solid figure bounded by a ''sphere''; it is also called a solid sphere. It may be a closed ball (including the boundary points that constitute the sphere) or an open ball (excluding them).
These concepts are defin ...
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 one-parameter 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
*
* {{MathSciNet, id=1333890 (See chapter 3)
Measures (measure theory)