Some remarks on countably determined measures and uniform distribution of sequences

Two topics are investigated: countably determined (regular Borel probability) measures on compact Hausdorff spaces, and uniform distribution of sequences regarding mainly this kind of measures. We prove several characterizations of countably determined measures, and apply the results in order to show the existence of a well distributed sequence in the support of a countably determined measure. We also generalize a result of Losert on the existence of uniformly distributed sequences in compact dyadic spaces.