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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- May 3rd 2010, 08:38 AMderek walcottexistance 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 - May 3rd 2010, 08:49 AMJG89
Oops, I missed the strictly positive part.

- May 3rd 2010, 10:11 AMJG89
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.