Lévy's Inversion Formula

Theorem
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.

Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.

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

Let $a < b$ be such that:
 * $\map \Pr {X = a} = \map \Pr {X = b} = 0$

Then:
 * $\ds \map \Pr {a < X \le b} = \lim _{T \to \infty} \int^T _{-T} \dfrac {e^{-ita} - e^{-itb} }{it} \map \phi t \rd t$