Stability of Pexiderized homogeneity almost everywhere

In the paper we examine stability of Pexiderized φ-homogeneity equation f (αx) = φ(α)g(x) almost everywhere. In particular we prove, that if (G, ·, 0) is a group with zero, (G,X) is a G-space, Y is a locally convex vector space over K ∈ {R,C} and for functions φ : G→K, f, g : X→Y the difference f (αx) −φ(α)g(x) is suitably bounded almost everywhere in G×X, then, under certain assumptions on f , φ, g, the function φ is almost everywhere in G equal to c φ, where c ∈ K \ {0} is a constant and φ : G→K a multiplicative function, the function g is almost everywhere in X equal to a φ-homogeneous function F : X→Y , and the difference f − cF in some sense bounded almost everywhere in X. From this result we derive the stability of Pexiderized multiplicativity almost everywhere. © 2007 Elsevier Inc. All rights reserved