I'm trying to follow the proof of the scaling property of the Fourier transform but I've run into a problem.
The Fourier transform is defined as $F[g(t)] = \int g(t) e^{-jwt}dt $ but a part of the proof of the scaling property claims we must have (1) $F[g(at)] = \int g(at)e^{-jwat}dt$. This seems to contradict the definition of the Fourier transform as this should be $F[g(at)] = \int g(at)e^{-jwat}a \space dt$ but every source of the proof I read says (1) is correct.
Therefore is it a more accurate statement to say that the definition of the Fourier Transform is :
$$F[g(f(t))] = \int g(f(t))e^{-jwf(t)} dt$$ where $f(t)$ is some function of $t$?
The Fourier transform of a function $g$ is defined by $F[g(t)](w) = \int g(t) e^{-jwt}dt$. This is the end of definition; there is no room to additionally define it to be something else if $g$ happens to be a composition of two functions. If $g(t) = f(2t)$, then the above definition already says what the transform is, we can't introduce a new clause there.
The author probably made a change of variable in the integral $\int g(t) e^{-jwt}dt$, letting $t=as$ and then relabeling $s$ as $t$ again.
$$F[g(t)](w) = \int g(at) e^{-ajwt}d(at) = a\int g(at) e^{-ajwt}dt$$ Compare the latter integral to $$ F[g(at)](aw) = \int g(at) e^{-ajwt}dt$$ and the conclusion follows: $$ F[g(at)](w) = a^{-1}F[g(t)](a^{-1}w) $$ The omission of the frequency argument is a possible source of confusion here.