On measures of uniformly distributed sequences and Benford's law (Q1123926)

The metric theory of uniform distribution of sequences is complemented by considering product measures with not necessarily identical factors. A necessary and sufficient condition is given under which a general product measure assigns the value one to the set of uniformly distributed sequences. For a stationary random product measure, almost all sequences are uniformly distributed with probability one. The discrepancy is estimated by \(N^{-1/2}\log^ 3N\) for sufficiently large N. Thus the metric predominance of uniformly distributed sequences is stated, and a further explanation for Benford's law is provided. The results can also be interpreted as estimates of the empirical distribution function for non-identical distributed samples.