"$f\ge 0$, $g>0$, $fg\in L^1(\mathbb{R})$ and $g\notin L^1(\mathbb{R})$" implies "$f$ is integrable over $[-r,r]$"?

Question

Let $f\ge 0$ and $g>0$ be such that $fg\in L^1(\mathbb{R})$ but $g\notin L^1(\mathbb{R})$. Can we get the following conclusion: $f$ is integrable over $[-r,r]$ for any $r>0$ ?

Intuitively I think this is true and I can't find any counter example at the moment. But I can't give a proof.

Answer 1

Let $f(x) = \frac{1}{|x|}$ and $g(x) = \chi_{|x|>1}\frac{1}{|x|}+\chi_{|x|\le 1}|x|$...

Note here that technically $g(0)=0$, but at least $g>0$ almost everywhere.