\begin{definition}[Definition:Characteristic Function of Random Variable]
Let $\struct {\Omega, \Sigma, \Pr}$ be a probability space.
Let $X$ be a real-valued random variable on $\struct {\Omega, \Sigma, \Pr}$.
The '''characteristic function of $X$''' is the mapping $\phi: \R \to \C$ defined by:
:$\map \phi t = \expect {e^{i t X} }$
where:
:$i$ is the imaginary unit
:$\expect \cdot$ denotes expectation.
\end{definition}