Let [n] = {1, 2, . . . , n} be the standard n-element set, 2 [n] its power set. The size |F| of a family F ⊂ 2 [n] is the number of subsets F ∈ F. In the present paper we consider the more equitable measurement F where the contribution of each F ∈ F is 1 n |F | . In particular, 2 [n] = n + 1. Our main result is that for every positive integer s, F > (n + 1)s/s + 1 implies the existence of s pairwise disjoint members in F. Moreover, this is best possible for all n ≥ s. The corresponding classical result of Kleitman is an easy consequence. The relatively simple proofs seem to indicate that F is the right measurement for such problems.