On families deciding Rosenthal's lemma

A classical version of Rosenthal's lemma reads as follows. Given a sequence (mu_k) of finitely additive bounded positive measures on P(N) and an infinite antichain (a_n) in P(N), for every epsilon > 0 there exists an infinite subset A of N such that for every k in A the following inequality is satisfied:

mu_k(U{a_n: n in A, n=/=k}) < epsilon.

During the talk I shall show that assuming the existence of selective ultrafilters the set A can be chosen from a basis of a given selective ultrafilter. This will yield some interesting applications in functional analysis.