the uniqueness theorem for characteristic functions
For the characteristic function to be trustworthy as a stand-in for the distribution, we need to know that no two different distributions can hide behind the same characteristic function. Just as a fingerprint is useless if two people can share one, an encoding of a distribution is useless if it is ambiguous. The uniqueness theorem guarantees there is no ambiguity: the characteristic function is a perfect fingerprint.
The statement is clean: if two random variables X and Y have the same characteristic function, phi_X(t) = phi_Y(t) for all real t, then X and Y have exactly the same distribution. Crucially, and unlike the mgf's uniqueness theorem, there is no extra condition to check — no requirement that anything be finite in a neighbourhood, no exceptions for heavy tails. Since the characteristic function always exists for every distribution, the uniqueness theorem applies to every distribution, full stop. This is the rigorous backbone that lets you argue 'I computed the characteristic function and it equals that of a Normal, therefore the variable is exactly Normal' with complete confidence.
The deeper reason this works is that the characteristic function is the Fourier transform of the distribution, and the Fourier transform is invertible — there is an inversion formula that reconstructs the distribution from phi_X. Uniqueness is really the statement that this reconstruction is unambiguous. This is what makes the transform method legitimate from start to finish: encode a distribution as its characteristic function, manipulate that function freely (multiply for sums, take limits for convergence), and decode the result, knowing the answer you read off is the one and only distribution it can be.
You compute the characteristic function of some standardized sum and find it equals e^(-t^2/2) for all t. Since e^(-t^2/2) is the characteristic function of the standard normal, the uniqueness theorem forces the conclusion that the sum's distribution IS the standard normal — exactly, not approximately.
Equal characteristic functions force equal distributions — with no extra existence condition, unlike the mgf.
This is strictly stronger than the mgf's uniqueness theorem because it carries no fine print: it holds for every distribution, including heavy-tailed ones with no mgf and no moments.