Let be a sequence of real numbers. For put
More generally, a sequence of points in a finite measure space is uniformly distributed with respect to a family of sets if
![$\displaystyle Z(N,\alpha,\beta)=\operatorname{card}\{n\in[1..N] : \alpha \leq (u_n \bmod 1)< \beta \}.$](http://images.planetmath.org:8080/cache/objects/5746/l2h/img3.png "$\displaystyle Z(N,\alpha,\beta)=\operatorname{card}\{n\in[1..N] : \alpha \leq (u_n \bmod 1)< \beta \}.$"){kind=small}
![$ [0,1]$](http://images.planetmath.org:8080/cache/objects/5746/l2h/img9.png "$ [0,1]$"){kind=small}
![$\displaystyle \lim_{N\to\infty} \frac{\operatorname{card}\{n\in[1..N] :u_n\in S\}}{N}=\frac{\mu(S)}{\mu(X)}$](http://images.planetmath.org:8080/cache/objects/5746/l2h/img14.png "$\displaystyle \lim_{N\to\infty} \frac{\operatorname{card}\{n\in[1..N] :u_n\in S\}}{N}=\frac{\mu(S)}{\mu(X)}$")
for every 