Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X : \Omega \to \mathbb{R}^N$ is a D4383: Random euclidean real number on $P$

The characteristic function of $X$ is the D992: Function \begin{equation} \mathbb{R}^N \to \mathbb{C}, \quad \xi \mapsto \mathbb{E}_{\mathbb{P}}(e^{i t \cdot X}) \end{equation}

Let $P = (\Omega, \mathcal{F}, \mathbb{P})$ be a D1159: Probability space such that

(i)	$X : \Omega \to \mathbb{R}^N$ is a D4383: Random euclidean real number on $P$

The characteristic function of $X$ is the D992: Function \begin{equation} \mathbb{R}^N \to \mathbb{C}, \quad \xi \mapsto \int_{\Omega} e^{i t \cdot X} \, d \mathbb{P} \end{equation}