I'm having some trouble trying to recover the probability mass function of a discrete random variable from its characteristic function.
I have seen that some continuous cases, you can recognize that the characteristic function is the inverse Fourier transform of the density function, so you can apply the Fourier transform again. The formula in that case is $$f(x) = \frac{1}{2\pi} \int_\mathbb{R} e^{-itx} \phi(t) \mathop{dt}.$$
However, in the discrete case, this integral does not make sense, as it is $$\frac{1}{2\pi} \int_\mathbb{R} e^{-itx} \sum_{n=1}^N P(X=x_n) e^{itx_n} \mathop{dt} = \frac{1}{2\pi} \sum_{n=1}^N P(X=x_n)\int_{\mathbb{R}}e^{it(x_n-x)} \mathop{dt},$$ which is not integrable. I thought about using an integral over an interval instead, but it is not possible to get the right period since the $x_n$ are arbitrary real numbers.
How do you recover the pmf from the characteristic function?