On a composite functional equation satisfied almost everywhere

The purpose of the paper is to deal with functions on an Abelian group (G,+)(G,+) which satisfy the following composite functional equation: f(f(x)+y−f(y))=f(x)f(f(x)+y−f(y))=f(x) for almost all pairs (x,y)∈G×G(x,y)∈G×G. The concept of “almost all pairs” is defined in an abstract manner. Our main result states that, under some technical assumptions, every such a function behaves in some sense similar to a function which satisfy the above equation for all pairs (x,y)∈G×G(x,y)∈G×G.