In the book "Classical Fourier Analysis by Grafakos" the Fourier transform is first defined on $S(\mathbb{R}^n)$. In the Chapter 2.2.4 it is mentioned that the definition of the Fourier transform $\hat{f}(\xi)=\int_{\mathbb{R}^n}e^{-2 \pi i x \xi} dx$ "makes sense" as a convergent integral for functions $f \in L_1(\mathbb{R}^n)$.
Here my question:
First I thought he meant that $\hat{f}(\xi)$ in $L_1$ can be approximated with a sequence. But I think I did misinterpret it. I did stumble over the following question: Showing existence of Fourier-Transform
Reading through it, I now think that the "makes sense as"-part in the book was meant as a "the proof for $S(\mathbb{R}^n)$ holds too for $L_1(\mathbb{R}^n)$". (In the Link only the case $n=1$ is discussed, but I think a very similar proof would do for $n \neq 1$)
Now my question: Did I now understand it correctly? The "makes sense as"-part did just mean that the same proof also yields the existence of the FT for $L_1$, right?