Families of sets related to Rosenthal's lemma (Q1712928)

A matrix of non-negative reals $\langle m^k_n:n,k\in\omega\rangle$ is said to be a Rosenthal matrix, if $\sum_{n\in\omega}m^k_n\le1$ for every $k\in\omega$. A family $\mathcal{F}\subseteq[\omega]^\omega$ is called Rosenthal if for every Rosenthal matrix $\langle m^k_n:n,k\in\omega\rangle$ and every $\varepsilon>0$ there exists $A\in\mathcal{F}$ such that for every $k\in\omega$, $\sum_{n\in A\setminus\{k\}}m^k_n<\varepsilon$. This notion was obtained by analysing the proof of Rosenthal's lemma setting $m^k_n=\mu_k(a_n)$ for an antichain $\langle a_n:n\in\omega\rangle$ and a bounded sequence of finitely additive non-negative measures $\langle\mu_k:k\in\omega\rangle$ in an arbitrary Boolean algebra. In this setting, Rosenthal's lemma states that $[\omega]^\omega$ is a Rosenthal family. The paper under the review solves the question whether a given family $\mathcal{F}$ is a Rosenthal family. The author proves the following results: The cardinality of a Rosenthal family cannot be less than the covering of the category $\text{cov}(\mathcal{M})$ and every base of a selective ultrafilter is a Rosenthal family. Under Martin's axiom for $\sigma$-centered partially ordered sets there exists a non-selective ultrafilter which is a Rosenthal family (in fact it is a P-point that is not a Q-point). The iterated Sacks forcing of length $\omega_2$ provides a model of ZFC in which there exists a Rosenthal family of cardinality $<\mathfrak{c}$.