Abstract

A family \ is called Rosenthal if for every Boolean algebra \, bounded sequence \ of measures on \, antichain \ in \, and \, there exists \ such that \<\varepsilon \) for every \. Well-known and important Rosenthal’s lemma states that \ is a Rosenthal family. In this paper we provide a necessary condition in terms of antichains in \}\) for a family to be Rosenthal which leads us to a conclusion that no Rosenthal family has cardinality strictly less than \\), the covering of category. We also study ultrafilters on \ which are Rosenthal families—we show that the class of Rosenthal ultrafilters contains all selective ultrafilters.