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A sequence s _{1}, s _{2}, … in U = [0, 1) is said to be uniformly distributed if, in the limit, the number of s_{j} falling in any given subinterval is proportional to its length. Equivalently, s _{1}, s _{2}, … is uniformly distributed if the sequence of equiweighted atomic probability measures ?_{N} (s_{j} ) = 1/N, supported by the initial N-segments s _{1}, s _{2}, …, s _{N}, converges weakly to Lebesgue measure on U. This notion immediately generalizes to any topological space with a corresponding probability measure on the Borel sets.
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