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In general, I know that it is not necessarily the case that the product of two Lebesgue integrable functions $f,g$ will be Lebesgue integrable. But I was reading in a textbook that if at least one of these functions is bounded, then their product will be Lebesgue integrable. How can we prove this statement? I'd appreciate some input on this, thanks.

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This is the easiest case of Hölder's inequality and it would have been as quick to write what Didier wrote as to fetch this link. Still, it's worthwhile to read this page. – t.b. Aug 6 '11 at 18:08
up vote 10 down vote accepted

If $|f|\le M$ almost everywhere, then $|fg|\le M|g|$ almost everywhere hence $\displaystyle\int|fg|\le M\int|g|$ is finite because $g$ is integrable.

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