Why does the characteristic function always exist?

Question

I've read that the characteristic function of a probability distribution always exists because it's bounded. However, the characteristic function is still Taylor expanded in terms of the moments of a given probability distribution. Given the the moments don't necessarily exist, why is it that the characteristic function still exists?

Answer 1

If $X$ is a random variable, then what it means for the characteristic function of $X$ to exist is that the random variable $|e^{itX}|$ must have finite expectation for every $t$. This is automatically true because $|e^{itX}|=1$ for all $t \in \mathbb{R}$, hence $E(|e^{itX}|)=1<\infty$. This is true regardless of whether or not the moments exist. The characteristic function can be Taylor expanded in terms of the moments only if the moments actually exist, but in general they won't.