I remember back in undergraduate to ask myself this question : In the general case, I is an interval, \int_I fg =! \int_I f \int_I g (*) But how to describe the egality case, i.e find all couples (f,g) of L^1 this case ?
closed as off topic by Scott Morrison♦ Oct 22 '09 at 2:51Questions on MathOverflow are expected to relate to research level mathematics within the scope defined by the community. Consider editing the question or leaving comments for improvement if you believe the question can be reworded to fit within the scope. Read more about reopening questions here. If this question can be reworded to fit the rules in the help center, please edit the question. 


If the measure of the domain of integration is 1, the identity \int_I fg = \int_I f \int_I g is equivalent to: \int_I (f\int_I f)(g\int_I g) = 0, which is an L^2 orthogonality condition, which I don't think has some equally "elementary" equivalent formulation. 


I'm not sure that the answer will be that informative. For instance, assuming your interval has length 1 (just to simplify the notation), one can take $g$ to be any constant and $f$ to be any integrable function. Or, take $f$ and $g$ to have disjoint supports, but to also have mean value $0$. That will work too. I can't at the moment see any obvious set of conditions which will be both necessary and sufficient. 

