On measurable solutions of a general functional equation on topological groups

Abstract

We establish a theorem of the type "measurability implies continuity" for solutions f of the functional equationunder reasonable conditions upon the integers αi, βi and the mappings Γ, Φ.

The stability of functional equation min{f(x + y), f(x - y)} = |f(x) - f(y)|

“…Baron in [3] showed that if G is a metrizable topological group and f : G ℝ is Baire measurable and satisfies (1.2) then f is continuous. Kochanek and Lewicki (see [4]) proved that if G is metrizable locally compact group and f : G ℝ is Haar measurable and satisfies (1.2), then f is continuous. As already mentioned in [2], Kochanek noticed that every function f defined on an abelian group G which is of the form f = g ∘ a, where g : ℝ ℝ is a solution of (1.2) described by Theorem 1.1 and a: G ℝ is an additive function, is a solution of…”

On Baire measurable solutions of some functional equations

“…Baron for (1). Lebesgue measurable solutions of (1), and in fact Haar measurable solutions of much more general equations, were considered by T. Kochanek and M. Lewicki in [3]. We consider Baire measurable solutions of…”