Originally Posted by

**rargh** Let x be a random real univariate variable described by a probability distribution d(x).

d(x) is generally a distribution, not a function, but we know it has a well defined measure on a sigma algebra and that it is always positive and the measure of the whole set is 1.

Does d(x) always have a Fourier Transform (or equivalently, a characteristic function)?

I think it does, but I'm not 100% sure because one might think of a not so well behaved probability distribution (distribution, not function) that might not be fourier transformable.

I think that we simply use the fact that, defining f(w) as E[exp(iwx)], we know from the basic property of an integral (or measure) that |f(w)|<=E[|exp(iwx)|]=E[1]=1