On Almost Everywhere K-Additive Set-Valued Maps

Abstract

Abstract Let X be an Abelian group, Y be a commutative monoid, K ⊂Y be a submonoid and F : X → 2 Y \ {∅} be a set-valued map. Under some additional assumptions on ideals ℐ 1 in X and ℐ 2 in X 2, we prove that if F is ℐ 2-almost everywhere K-additive, then there exists a unique up to K K-additive set-valued map G : X → 2 Y \{∅} such that F = G ℐ 1-almost everywhere in X. Our considerations refers to the well known de Bruijn’s result [1].