Rosenthal families, pavings, and generic cardinal invariants

Abstract

Following D. Sobota we call a family F \mathcal F of infinite subsets of N \mathbb {N} a Rosenthal family if it can replace the family of all infinite subsets of N \mathbb {N} in the classical Rosenthal 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 r \mathfrak r . This is achieved through analyzing nowhere reaping families of subsets of N \mathbb {N} and through applying a paving lemma which is a consequence of a paving lemma concerning linear operators on ℓ 1 n \ell _1^n due to Bourgain. We use connections of the above results with free set results for functions on N \mathbb {N} and with linear operators on c 0 c_0 to determine the values of several other derived cardinal invariants.