A (finitely additive) measure \(\mu\) on a Boolean algebra \(\mathfrak B\) is said to be separable if there is a countable \(\mathfrak A\subseteq\mathfrak B\) such that for all \(\varepsilon>0\) and \(b\in\mathfrak B\) there is \(a\in\mathfrak A\) with \(\mu(a\bigtriangleup b)<\varepsilon\). A compact space \(K\) is said to be approximable if there is a sequence of probability measures \(\langle\mu_n:n\in\omega\rangle\) on \(K\) such that for every open set \(O\subseteq K\) there is \(n\) such that \(\mu_n(O)>1/2\).
A motivation for the paper is to find a combinatorial characterization of Boolean algebras which carry a strictly positive separable measure. M. Talagrand [“Séparabilité vague dans l’espace des mesures sur un compact”, Isr. J. Math. 37, 171–180 (1980; Zbl 0445.46022)] found under CH an example of a Boolean algebra \(\mathfrak B\) such that the Stone space of \(\mathfrak B\) is approximable and \(\mathfrak B\) does not support a separable strictly positive measure.
The authors in the paper under the review present an example of such a Boolean algebra \(\mathfrak B\) of size \(\mathfrak c\) in ZFC. For every \(\sigma\)-finite cc Boolean algebra they construct a ccc forcing which makes it have a strictly positive measure and prove that under \(\text{MA}+\neg\text{CH}\) every atomless ccc Boolean algebra of size \(<\mathfrak c\) carries a nonatomic separable strictly positive measure. Finally they obtain a combinatorial characterization of Boolean algebras (a kind of a chain condition) that carry a strictly positive nonatomic measure.