An alternative Cauchy equation almost everywhere

Abstract

Let (S, +) be a commutative semigroup and let ζ be a proper left translation invariant σ-ideal in S. Suppose that f : S → R satisfies |f(x + y)| = |f(x) + f(y)| Ω(ζ)-almost everywhere in S x S. We prove that there exists, exactly one homomorphism α : S → R such that α = f ζ-almost, everywhere in S.