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.
I'm stuck on the following question:
If , are -measurable functions in a measure space such that and are -integrable, show that the product is -integrable.
My first instinct was to apply this theorem (which my textbook doesn't give a name):
A function is -integrable iff is -measurable and there exists an integrable dominant for
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.