To keep things simple, let $X$ be a random variable, $F$ its distribution function, and $\phi$ its characteristic function. If $0$ is a continuity point of $F$, then $$ F(0)=\frac{1}{2}+\frac{i}{2\pi}\int_0^{\infty}\frac{\phi(t)-\phi(-t)}{t}dt $$ Are there conditions on $X$ that ensure that this integral is Lebesgue-integrable? (i.e. we can put absolute values around the integrand and still have a finite value)
2026-04-27 14:09:42.1777298982
in the Fourier inverse formula for a distribution function, the integral is Lebesgue-integrable?
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Unfortunately, I do not believe that necessary and sufficient conditions for the absolute convergence of this integral are currently known.
Your inversion formula is a special case of the Gil-Pelaez theorem, I found in this
this paper
which references Gil-Pilazez's paper
However, I an unable to find an analytical result, only papers regarding numerical approximations.