Glasnik Matematicki, Vol. 36, No.1 (2001), 1-9.

ON FUNCTIONS WHICH ARE ALMOST ADDITIVE MODULO A SUBGROUP

Janusz Brzdek

Abstract.   Let (X,+) be a commutative semigroup, uniquely divisible by 2, (G,+) be a topological group, and K be a discrete, normal and contable subgroup of G. We show that if X is endowed with a topology and the topologies in X and G satisfy some additional conditions, then for every measurable function f mapping X into G such that f(x+y) - f(x) - f(y) ∈ K almost everywhere in X², with respect to some ideal in X², there is an additive function A : X → G with f(x) - A(x) ∈ K almost everywhere in X.

1991 Mathematics Subject Classification.   39B52.

Key words and phrases.   Cauchy difference, measurability, additive function, σ-ideal.