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;''.

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| author-link = Pertti Mattila

| title = Geometry of sets and measures in Euclidean spaces: Fractals and rectifiability

| url = https://archive.org/details/geometryofsetsme0000matt

| url-access = registration

| series = Cambridge Studies in Advanced Mathematics No. 44

| publisher = Cambridge University Press