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≠kμk(an)<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).