I asked myself a question during my first year and never find any interesting clue about it.
One of the first thing you learned in integral calculus is that $\int fg \neq \int f \int g$ in general. One can ask the following question : When do we have $\int_I fg = \int_I f \int_I g$ ? Can I find a maximal subset of $CM(K)^2$ or $L^1(\mathbb{R})^2$ for example ? A full "algebric" characterization of such couples ? Sure there are some trivial couples, but the more the time passes, the more I have the feeling that the answer is : this set exists, and there is nothing more interesting to say about it. But it is not really a big satisfaction ... Any ideas/answers about these questions ?
PS : It's my first post on math.se, and it seems that my questions are appropriate after reading the FAQ. I hope so, but please redirect me if you think that I'm not on the right place.