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I can't find in any book on measure and integral whether a product of two Lebesgue integrable functions is also integrable. Could someone clarify for me under what circumstances it is true?

To be specific, I have an assumption $fg\in L_{\mathrm{loc}}^1 (\Omega)$ and I want to determine what requirements on $f$ a $g$ are sufficient to hold this assumption.

vessel
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  • The most standard assumption that will imply this is that $f$ and $g$ are both in $L_{loc}^2(\Omega)$. You can prove the sufficiency of this assumption by applying Cauchy-Schwarz to $fg1_K$ where $K$ is a compact set. – Chris Janjigian Apr 24 '13 at 17:33
  • @ChrisJanjigian Thank you, that should help. Could you give me an advice what to do if I have the assumption $fg\in L^{1+\eta}(\Omega), \eta>0$? I cannot use the Hölder's inequality now, can I? – vessel Apr 25 '13 at 07:42

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