1. ## integrable functions

I'm stuck on the following question:

If $\displaystyle f$, $\displaystyle g$ are $\displaystyle \mu$-measurable functions in a measure space $\displaystyle (S, \Sigma, \mu)$ such that $\displaystyle |f|^2$ and $\displaystyle |g|^2$ are $\displaystyle \mu$-integrable, show that the product $\displaystyle fg$ is $\displaystyle \mu$-integrable.

My first instinct was to apply this theorem (which my textbook doesn't give a name):
A function $\displaystyle f$ is $\displaystyle \mu$-integrable iff $\displaystyle f$ is $\displaystyle \mu$-measurable and there exists an integrable dominant for $\displaystyle f$
But I can't see any way to get the integrable dominant, although the measurable part is easy. Most of the previous exam questions for this subject were very easy so I think I might be missing something quite basic.

Thanks for bothering to read up to here, even if you can't help.

2. Hello,

For me, the first theorem that comes in mind is Cauchy-Schwarz inequality
and it works, considering that a function is integrable if its integral is finite.