A criterion for the uniform distribution of sequences in compact metric spaces

Let X be a compact metric space, le μ be a non-negative normalized Borel measure on X and let f be a measurable bounded real-valued function defined on X such that f is μ-almost everywhere continuous and different from zero. It is proved that a sequence (x _n ), n=1,2, ... of points in X is μ-uniformly distributed if and only if for every Borel set E⊆X with μ(Bd(E))=0 we have {Mathematical expression} where 1 _E denotes the characteristic function of E and bdE the boundary of E. Furthermore some quantitative aspects and generalizations of this theorem are discussed.