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.:
 * $\forall A \in \map \BB \R : \map {m_T} A := \map m { A \cap \closedint {-T} T}$

Let $\mu \times m_T$ be the product measure.

Then:

where:
 * $\ds \map f {x,t} := \dfrac {e^{it \paren {x - a} } - e^{it \paren {x - b} } }{it}$

The $f$ is essentially bounded with respect to $\mu \times m_T$, since:
 * $\forall \struct {x, t} \in \R \times \R_{\ne 0} : \cmod {\dfrac {e^{it \paren {x - a} } - e^{it \paren {x - b} } }{it} } \le \dfrac 3 2$

by Bounds for Complex Exponential.

In particular, $f$ is $\mu \times m_T$-integrable, as $\mu \times m_T$ is finite.

Thus by Fubini's Theorem: