Stability of Pexiderized homogeneity almost everywhere

In the paper we examine stability of Pexiderized ϕ-homogeneity equationf(αx)=ϕ(α)g(x)f(αx)=ϕ(α)g(x) almost everywhere. In particular we prove, that if (G,⋅,0)(G,⋅,0) is a group with zero, (G,X)(G,X) is a G-space, Y   is a locally convex vector space over K∈{R,C}K∈{R,C} and for functions ϕ:G→Kϕ:G→K, f,g:X→Yf,g:X→Y the differencef(αx)−ϕ(α)g(x)f(αx)−ϕ(α)g(x) is suitably bounded almost everywhere in G×XG×X, then, under certain assumptions on f, ϕ, g, the function ϕ is almost everywhere in G   equal to cϕ˜, where c∈K∖{0}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→YF:X→Y, and the difference f−cFf−cF in some sense bounded almost everywhere in X. From this result we derive the stability of Pexiderized multiplicativity almost everywhere.