[Math] Product of two Lebesgue integrable functions

Question

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.

Answer 1

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.