Rosenthal families, pavings and generic cardinal invariants

Abstract

Following D. Sobota we call a family $\mathcal F$ of infinite subsets of $\mathbb N$ a Rosenthal family if it can replace the family of all infinite subsets of $\mathbb N$ in classical Rosenthal's Lemma concerning sequences of measures on pairwise disjoint sets. We resolve two problems on Rosenthal families: every ultrafilter is a Rosenthal family and the minimal size of a Rosenthal family is exactly equal to the reaping cardinal $\mathfrak r$. This is achieved through analyzing nowhere reaping families of subsets of $\mathbb N$ and through applying a paving lemma which is a consequence of a paving lemma concerning linear operators on $\ell_1^n$ due to Bourgain. We use connections of the above results with free set results for functions on $\mathbb N$ and with linear operators on $c_0$ to determine the values of several other derived cardinal invariants.