Some remarks on countably determined measures and uniform distribution of sequences (Q1909222)

Let \(X\) be a compact Hausdorff space. A regular Borel probability measure \(\mu\) on \(X\) is called countably determined if there exists a sequence \((U_n)\) of Borel sets in \(X\) such that, for every compact subset \(K\) of \(X\), \(\mu(K)= \inf\{\mu(U_n): U_n\subseteq K\}\). If \((U_n)\) can be chosen as open \(F_\sigma\)-sets then \(\mu\) is called uniformly regular. These notions go back to \textit{R. Pol} [Pac. J. Math. 100, 185-201 (1982; Zbl 0522.46019)]. In the present paper, the author studies a number of properties of these classes of measures and discusses various examples. A typical result (Theorem 1.7 (iii)): \(\mu\) is uniformly regular iff there exist a compact metric space \(M\) and a continuous surjection \(\varphi: X\to M\) such that for every \(\mu\)-Jordan measurable set \(B\subseteq X\), the image \(\varphi(B)\) is \(\lambda\)-Jordan measurable and \(\mu(B)= \lambda(\varphi(B))\), where \(\lambda= \varphi(\mu)\). The second part of the paper deals with applications to results on uniform distribution. For example, Theorem 2.9 says that in the case of a uniformly regular measure \(\mu\), any sequence \((x_n)\) for which every open non-empty subset of \(X\) contains infinitely many terms has a well distributed rearrangement. The last Theorem 2.17 generalizes the reviewer's result on the existence of uniformly distributed sequences in compact dyadic spaces of topological weight \(\leq\omega^+\).