A criterion for the uniform distribution of sequences in compact metric spaces
The author generalizes a result of \textit{J. Horbowicz} [Indagationes Math. 43, 301-307 (1981; Zbl 0465.10039)] to compact metric probability spaces (X,\(\mu)\): Let f be a bounded measurable function such that f is \(\mu\)- almost everywhere continuous and different from zero. Then the sequence \((x_ n)\) in X is \(\mu\)-uniformly distributed if and only if \[ \lim_{N\to \infty}(1/N)\sum_{n\leq N, x_ n\in E}f(x_ n)=\int_{E}f(x) dx \] for every Jordan-measurable subset E of X (i.e. the boundary of E has measure zero). The result is generalized to more general summation methods and a situation in which \(\int_{E}f(x) dx\) is replaced by \(\int_{X}f(x)(Tg)(x) d\mu\), T: \(B\to C(X)\) a compact operator, such that C(X) is the closed linear span of T(B) and the constant functions.
scientific article; zbMATH DE number 4069004