TITLE: Characteristic Functions and motivations
QUESTION [8 upvotes]: 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.

REPLY [7 votes]: Practically speaking, the short answer is that it's convenient.  The characteristic function has better analytic properties than the moment generating function, lets you study all of the moments of a random variable at once, and has the extremely convenient property that $\phi_{X+Y}(t) = \phi_X(t) \phi_Y(t)$ if $X, Y$ are independent.  This makes the characteristic function an amazing tool for understanding sums of independent random variables, and indeed a standard proof of the central limit theorem proceeds via a computation of characteristic functions.  
Many constructions in mathematics translate problems in one domain (understanding distribution functions) to problems in another domain (understanding characteristic functions), and these constructions are useful because different tools apply in the second domain.  That is exactly what happens in the Fourier-theoretic proof of the central limit theorem.