Recently, I encountered some questions regarding the characteristic function and how it determines the distribution uniquely. Say I have a random variable $X$ with a standard gaussian distribution. Let's consider an "approximating" discrete random variable $\tilde{X}$ to $X$. $\tilde{X}$ has support on $\frac{i}{n}$ for $i \in \text{Z}$ and fixed $n$. The density of $\tilde{X}$ take value $\frac{1}{2 \pi n} e^{-i^2/2n^2} / S$ on $\frac{i}{n}$. (S is the sum of probability at each point so that it normalize to $1$.) Intuitively, the density of $\tilde{X}$ and $X$ look like the following:
We know the Fourier inversion of characteristic function uniquely determined the distribution (see e.g. Note on the inversion theorem). Moreover, the difference between the distributions of $X$ and $\tilde{X}$ can be arbitrarily small for large $n$. However, the characteristic function of $\tilde{X}$ is periodic (since it has support on the lattice), while the characteristic function of $X$ is exponential decay. So, my question is, why can this happen? The distributions are "close," but the characteristic functions are very different. Does the Fourier inversion cancel out the periodicity so that it does not affect the distribution too much? Thank you so much! This is more like a question about intuitions.