Uniqueness theorem for characteristic functions

Two finite Borel measures on $\mathbb{R}$ with the same characteristic function are equal. One proof convolves both measures with centered Gaussians, applies Fourier inversion to the smooth densities, and then lets the Gaussian variance tend to zero.

Let $f$ be compactly supported and continuous, with Fourier convention $\hat{f}(t)=\int f(x)e^{itx}dx$. Let $g,\hat{g}\in L^1(\mathbb{R})$, choosing $g$ as its continuous Fourier-inversion representative, and suppose $g(0)\ne 0$. ThenFubini theorem rewrites the integral as an average of translates of $f$ against the rescaled integrable kernel $\hat{g}$, whose total mass is $2\pi g(0)$. Dominated convergence gives the limit. Requiring the continuous representative matters: arbitrarily changing the value $g(0)$ does not change its Fourier transform but would incorrectly change the denominator. The formula shows that a characteristic function determines every compactly supported continuous test expectation.