Is the product of a Schwartz function and a locally integrable function integrable?

Question

Let $f\in S(\mathbb{R}^n)$ the space of rapidly decreasing functions on $\mathbb{R}^n$ and $g\in L_{loc}^1(\mathbb{R}^n)$. Is $fg$ integrable? Namely is it true that $$ \int |fg| <\infty. $$

Answer 1

No, pick $n=1$, $f(x)=\xi(x)e^{-|x|}$ and $g(x)=e^{x^2}$, where $\xi \equiv 0$ near $x=0$. But even $f(x)=e^{-x^2}$, as Daniel Fischer points out in a comment.

Here the point is that any continuous function is locally integrable, and $f$ cannot compete with functions that blow up at infinity faster than polynomials.