Some remarks on countably determined measures and uniform distribution of sequences

Abstract

Abstract. Two topics are investigated: countably determined (regular Borel probability) measures on compact Hausdorffspaces, 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.

On Corson compacta and embeddings of 𝐶(𝐾) spaces

“…for every open U ⊆ K; see Pol [24] and Mercourakis [18]. It will be convenient to say, whenever ( †) holds, that F approximates U from below (with respect to some μ ∈ P (K), which is clear from the context).…”

“…This is clear in case (i), since every μ ∈ P (K) on a scattered compactum is purely atomic (concentrated on a countable set). If K is linearly ordered, then every μ is again countably determined by a result to be found in [18].…”

Extension operators and twisted sums of c0 and C(K) spaces

“…It follows that the sequence (u f q n x ) n∈ω converges to u f y,f (x) in ult(A f ). The space M(I B ) has the cardinality c, which follows for instance from the fact that every probability measure on I B has a uniformly distributed sequence, see Mercourakis [24]. Hence the family E of all maps e : Q → M(I B ) has the same cardinality.…”

On sequential properties of Banach spaces, spaces of measures and densities

“…Part (ii) follows from Theorem 5.1 since the algebra A, as a tree algebra, is minimally generated, see e.g. [4] (actually, S(K) = P (K) can be also derived from a result due to Sapounakis that every measure on K has a uniformly distributed sequence, see [20]). …”

“…Mercourakis [20] mentions several classes of compact spaces K for which every µ ∈ P (K) admits a uniformly distributed sequence.…”