# Biggest subset of $L^1$ for which products and integrals commute

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.

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I don't really understand what your title has to do with your question. I also don't think this is an interesting condition to consider. – Qiaochu Yuan Feb 3 '11 at 22:34
Ouch. I did not state correctly the question. When $\int_I fg = \int_I f \int_I g$. Of course, the negation is not really interesting. – Sam Feb 3 '11 at 22:46
Title suggestion: Maybe "biggest subset of $L^1$ for which products and integrals commute"? – Jesse Madnick Mar 6 '11 at 23:47
Indeed. Thanks for the advice. – Sam Mar 6 '11 at 23:49

I lied when I said that this condition is not interesting. If $f, g$ are random variables and $I$ a probability space, then this is condition is satisfied if $f, g$ are independent. So that's an interesting source of examples.