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Math Help - existance of an integrable product of fg

  1. #1
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    Thumbs up existance of an integrable product of fg

    Does there exist a strictly positive non-integrable functions f,g:[0,1]-->R such that their product fg is integrable?
    Please provide an example or prove that they don't exist.

    just a question i was pondering
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  2. #2
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    Oops, I missed the strictly positive part.
    Last edited by JG89; May 3rd 2010 at 11:07 AM.
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  3. #3
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    Let f(x) = 1/2 if x is rational and be equal to 2 if x is irrational. Let g(x) = 2 if x is rational and be equal to 1/2 if x is irrational. Both f and g are not Riemann-integrable (I'm assuming this is the type of integrability you're talking about) in [0,1] since they both have an uncountable amount of discontinuities in the interval. But fg = 1, which is integrable.
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