Stability of homogeneity almost everywhere

We consider approximately h-homogeneous mappings almost everywhere, that is functions F such that the difference F(ax) - h(a)F(x) is in some sense bounded almost everywhere in a product space. We will prove, under some assumptions, that then either we have some kind of boundedness of h and F, or there exist a homomorphism [Equation] and a [Equation]-homogeneous function [Equation], which are almost everywhere equal to h and F, respectively. From this result we derive the superstability effect for the multiplicativity almost everywhere.