$f\cdot g$ is integrable, g is integrable, can we deduce that f is integrable?

Question

$f\cdot g$ is Lebesgue integrable, g is Lebesgue integrable, can we deduce that f is Lebesgue integrable?

$f\cdot g$ is integrable, g is integrable, can we deduce that f is finite a.e.?

Answer 1

If the support of $g$ and the support of $f$ have intersection with measure zero, you can't conclude anything.

Answer 2

For a nontrivial example, take $f=\frac{1}{x}\chi_{[1,+\infty)}$ and $g=\frac{1}{x^2}\chi_{[1,+\infty)}$. Then $fg$ is integrable, $g$ is integrable, but $f$ is not.

As for the second question, in general you can't conclude anything, as pointed out already.