Lévy's Inversion Formula for Integrable Characteristic Function

Theorem
Let $X$ be a real-valued random variable.

Let $P_X$ be the probability distribution of $X$.

Let $\phi : \R \to \C$ be the characteristic function of $X$.

Suppose that $\phi$ is Lebesgue integrable, i.e.:
 * $\ds \int _\R \cmod {\map \phi t} \rd t < + \infty$

Then $P_X$ is absolutely continuous with respect to the Lebesgue measure.

Moreover, the Radon-Nikodym derivative is give by:
 * $\ds \map g x := \dfrac 1 {2 \pi} \int \map \phi t e^{- i t x} \rd t$