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:
 * $\ds \map \Pr {a < X \le b} = \lim _{T \to \infty} \frac{1}{2 \pi} \int^T _{-T} \dfrac {e^{-ita} - e^{-itb} }{it} \map \phi t \rd t$

Proof
Let $\mu$ be the probability distribution of $X$.

Let $m$ be the Lebesgue measure on $\R$.

For $T > 0$ let $m_T$ be the restriction of $m$ to $\closedint {-T} T$, i.e.:
 * $\forall A \in \map \BB \R : \map {m_T} A := \map m { A \cap \closedint {-T} T}$

Then: