Quotient of two $L^1$ functions

Question

Suppose that we have two $L^1(\mathbb{R})$ functions, $f$ and $g$, where $g$ is non-zero almost everywhere. Can we conclude that their quotient $\frac{f}{g}$ is finite almost everywhere?

Answer 1

Yes, because $g \in \rm L^1(\mathbb R)$ is an equivalence class of functions, if it is non-zero almost everywhere, you can choose a function $\tilde{g} \in \mathcal L^1(\mathbb R)$ that is nowhere zero and such that the class of $\tilde{g}$ in $\rm L^1(\mathbb R)$ is equal to $g$.

Then you can define the element $\frac{1}{g}$ in $\rm L^1(\mathbb R)$ to be the class of $\frac{1}{\tilde g}$.