Families of sets related to Rosenthal's lemma

A family F⊆ [ω] ^ω is called Rosenthal if for every Boolean algebra A, bounded sequence 〈μk:k∈ω〉 of measures on A, antichain 〈an:n∈ω〉 in A, and ε> 0 , there exists A∈ F such that ∑ _{n ∈ A , n ≠ k}μ _k(a _n) < ε for every k∈ A. Well-known and important Rosenthal’s lemma states that [ω] ^ω is a Rosenthal family. In this paper we provide a necessary condition in terms of antichains in ℘(ω) for a family to be Rosenthal which leads us to a conclusion that no Rosenthal family has cardinality strictly less than cov (M) , the covering of category. We also study ultrafilters on ω which are Rosenthal families—we show that the class of Rosenthal ultrafilters contains all selective ultrafilters (and consistently selective ultrafilters comprise a proper subclass).