Abstract
LetX be a compact metric space, le μ be a non-negative normalized Borel measure onX and letf be a measurable bounded real-valued function defined onX such thatf is μ-almost everywhere continuous and different from zero. It is proved that a sequence (x n ),n=1,2, … of points inX is μ-uniformly distributed if and only if for every Borel setE⊆X with μ(Bd(E))=0 we have\(\mathop {\lim }\limits_{N \to \infty } \frac{1}{N}\sum\limits_{n = 1}^N {f(x_n )} 1_E (x_n ) = \int\limits_E {f(x)d\mu (x)} ,\) where 1 E denotes the characteristic function ofE andbdE the boundary ofE. Furthermore some quantitative aspects and generalizations of this theorem are discussed.
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Tichy, R.F. A criterion for the uniform distribution of sequences in compact metric spaces. Rend. Circ. Mat. Palermo 36, 332–342 (1987). https://doi.org/10.1007/BF02843743
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DOI: https://doi.org/10.1007/BF02843743