On the structure of measurable filters on a countable set (Q1194671)

For a sequence \(\hat{p}= \{ p_{n}: n \in \omega \}\) of reals such that \(0<p_{n}\leq 1/2\), let \(\mu_{\hat{p}}\) be the product measure on \(2^{\omega}\) such that \(\mu_{\hat{p}}(\{ x \in 2^{\omega}: x(n)=1\} )=p_{n}\) and \(\mu_{\hat{p}}(\{ x \in 2^{\omega}: x(n)=0\} )=1- p_{n}\) for \(n\in \omega\). The following combinatorial characterization of nonprincipial \(\mu_{\hat{p}}\)-measurable filters \({\mathcal F}\) on \(\omega\) (identified with the sets of characteristic functions of its elements) is the main result of the paper: there exists a family \(\{ {\mathcal A}_{n}: n\in \omega \}\) such that: (a) \({\mathcal A}_{n}\) consists a finitely many finite subsets of \(\omega\) for all \(n\in \omega\), (b) \(\bigcup {\mathcal A}_{n} \cap \bigcup {\mathcal A}_{m} = \emptyset\) whenever \(n \neq m\), (c) \(\Sigma_{n=1}^{\infty}\mu_{\hat{p}} (\{ X\subset \omega : \exists a\in {\mathcal A}_{n}: a\subset X \}) < \infty\), (d) \(\forall x \in {\mathcal F} \exists^{\infty} n \exists a \in {\mathcal A}_{n} (a \subset X)\). (An analogous characterization of filters with the Baire property was given by \textit{M. Talagrand} [Stud. Math. 67, No. 1, 13-43 (1980; Zbl 0435.46023)].) Moreover the paper contains some interesting results concerning intersections of \(\mu_{\hat{p}}\)-nonmeasurable filters and filters which are both null and meager.