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Thread: integrable functions

  1. #1
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    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.
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  2. #2
    Moo
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    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.
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