Characteristic function fourier stieltjes convention

Characteristic function of a real random variable uses ${\phi }_{\mu }(t)=\int e^{itx}\mu (dx)$. The convention in Fourier transform of a finite complex Borel measure is $\widehat{\mu }(\xi )=\int e^{-2\pi ix\xi }\mu (dx)\mathrm{.}$ For a Borel probability law, substitution of $\xi =-t\mathrm{/}(2\pi )$ gives identical integrands and hence ${\phi }_{\mu }(t)=\widehat{\mu }(-\frac{t}{2\pi }),\widehat{\mu }(\xi )={\phi }_{\mu }(-2\pi \xi )\mathrm{.}$ Both integrals exist because the integrand has modulus one and the law has mass one. In particular the substitution is a bijection of the real frequency line; equality of characteristic functions is exactly equality of the transforms in this convention. The identity at zero is $1=1$.

This is an identification of conventions. The finite-complex-measure theorem states countable choice for its total-variation machinery; the displayed probability-law identity uses only the already defined bounded probability integral. Consumers invoking Fourier uniqueness must retain the separate AC hypothesis of that uniqueness theorem.