Let $(X,\scr{M},\mu)$ be a measure space. Let $f,g:(X,\scr{M},\mu)\rightarrow[0,\infty]$ be measurable with finite integrals over $X$. Must $\int_X fg \,d\mu$ be finite ?
My guess is yes, I already tried proving it for sometime. I want to see if there is a counterexample . If there aren't any please inform me that the fact is true without saying a proof and I will try to prove it again, as I think it should be easy to prove if it's true.
Thank you
The question is equivalent to asking if $\int_X f^2\,d\mu$ must be finite. One of the directions is trivial, for the other direction just note that $$\int_Xfg\,d\mu=\frac{1}{2}[\int_X(f+g)^2\,d\mu-\int_Xf^2\,d\mu-\int_Xg^2\,d\mu]$$
The question can also be asked as : Is $L^1(\mu)$ a ring under pointwise multiplication and addition ?