## 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.