 Fourier-cutoff recovery of expectations from a characteristic function

Let $f$ be compactly supported and continuous, let $g,\hat{g}\in L^1(\mathbb{R})$, and use $\hat{g}(t)=\int g(x)e^{itx}dx$. If $g(0)\ne 0$, thenThis follows by Fubini's theorem, a change of variables, Fourier inversion theorem, and dominated convergence theorem. It gives a direct proof of the uniqueness theorem for characteristic functions because the right side depends only on ${\phi }_Z$.