On Almost Everywhere K-Additive Set-Valued Maps

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 ℐ₁ in X and ℐ₂ in X², 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 ℐ₁-almost everywhere in X. Our considerations refers to the well known de Bruijn’s result [1].