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Math Help - integrable functions

  1. #1
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    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.
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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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