A combinatorial theorem

AbstractLet 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 {Aξ, ξ < k} of pairwise-disjoint subsets of a we have α = ∩ξ < kΦ(Aξ). 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