I've recently studied characteristic functions in my probability course and I can't get why we define it to be the Fourier transform of the distribution (if the random variabile is continuous).
I mean that if $X$ is a random variable, $\varphi_X (t) = \mathbb{E}(e^{i t X}) = \int_{-\infty}^{+\infty} e^{i t x}f_X(x) dx$ where $f_X(x)$ is the distrubution function of $X$, and I can't see any motivation for doing this. I asked my professor but he wasn't clear at all; he said something like this:
"Since we proved the theorem that if $\varphi_X (t) = \varphi_Y (t)$ then $X \sim Y$ (or $P_X \equiv P_Y)$, it is natural to define it this way".
But of course, to prove that we need the definition! So I couldn't really make up my mind about it, if you could provide some help in this sense (motivation for defining the characteristic function of a random variable as the Fourier transform of its distribution) it would be much appreciated.