# integrable functions

• Nov 15th 2009, 09:36 PM
integrable functions
I'm stuck on the following question:

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

My first instinct was to apply this theorem (which my textbook doesn't give a name):
A function $f$ is $\mu$-integrable iff $f$ is $\mu$-measurable and there exists an integrable dominant for $f$
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.
• Nov 15th 2009, 10:17 PM
Moo
Hello,

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