“Pexiderized” homogeneity almost everywhere

In the paper we examine Pexiderized ϕ-homogeneity equation almost everywhere. Assume that G and H   are groups with zero, (X,G)(X,G) and (Y,H)(Y,H) are a G- and an H  -space, respectively. We prove, under some assumption on (Y,H)(Y,H), that if functions ϕ:G→H and F1,F2:X→Y satisfy Pexiderized ϕ-homogeneity equationF1(αx)=ϕ(α)F2(x)F1(αx)=ϕ(α)F2(x) almost everywhere in G×XG×X then either ϕ=0ϕ=0 almost everywhere in G   or F2=θF2=θ almost everywhere in X   or there exists a homomorphism ϕ˜:G→H such that ϕ=aϕ˜ almost everywhere in G   and there exists a function F¯:X→Y such thatF¯(αx)=ϕ˜(α)F¯(x)for α∈G∖{0},x∈X, andF1=aF¯almost everywhere in X,F2=F¯almost everywhere in X, where a∈H∗a∈H∗ is a constant. From this result we derive solution of the classical Pexider equation almost everywhere.