A combinatorial theorem

Abstract

Let a be an infinite cardinal, let k be a cardinal with k < α, and denote by P (α) the power set of α. The main result is the following: Let Φ: P (α) → P (α) be a function such that (i) Φ( A ∩ B) = Φ( A) ∪ Φ( B) for A, B ∈ P (α), (ii) for any family { A ξ , ξ < k} of pairwise-disjoint subsets of a we have α = ∩ ξ < k Φ( A ξ ). Then there is Γ ⊂ α, such that |Γ| = α and ξ ϵ Φ( Γ {ξ{ ) for all ξ ϵ Γ. A consequence of this theorem is another theorem concerning finite additive measures on a set, whose special cases are Rosenthal's result and Hajnal's theorem.