ABSTRACT
A sequence s 1 , s 2 , … $ s_1,s_2,\ldots $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math13_1.tif"/> in U = [ 0 , 1 ) $ { \text{ U}}=[0,1) $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math13_2.tif"/> is said to be uniformly distributed if, in the limit, the number of s j $ s_j $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math13_3.tif"/> falling in any given subinterval is proportional to its length. Equivalently, s 1 , s 2 , … $ s_1,s_2,\ldots $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math13_4.tif"/> is uniformly distributed if the sequence of equiweighted atomic probability measures μ N ( s j ) = 1 / N $ \mu _N(s_j)=1/N $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math13_5.tif"/> , supported by the initial N-segments s 1 , s 2 , … , s N $ s_1,s_2,\ldots ,s_N $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math13_6.tif"/> , converges weakly to Lebesgue measure on U $ {{ \text{ U}}} $ https://s3-euw1-ap-pe-df-pch-content-public-p.s3.eu-west-1.amazonaws.com/9781315119601/fb8178cb-c53c-4311-b072-eff5ec016aba/content/inline-math13_7.tif"/> . This notion immediately generalizes to any topological space with a corresponding probability measure on the Borel sets.
