Let $P = (\Omega, \mathcal{F}_{\Omega}, \mathbb{P})$ be a D1159: Probability space.
Let $M = (\Xi, \mathcal{F}_{\Xi})$ be a D1108: Measurable space.
A D18: Map $X : \Omega \to \Xi$ is a random variable from $P$ to $M$ if and only if \begin{equation} \forall \, E \in \mathcal{F}_{\Xi} : X^{-1}(E) \in \mathcal{F}_{\Omega} \end{equation}

Let $P = (\Omega, \mathcal{F}_{\Omega}, \mathbb{P})$ be a D1159: Probability space.
Let $M = (\Xi, \mathcal{F}_{\Xi})$ be a D1108: Measurable space.
A D18: Map $X : \Omega \to \Xi$ is a random variable from $P$ to $M$ if and only if \begin{equation} \forall \, E \in \mathcal{F}_{\Xi} : \{ X \in E \} \in \mathcal{F}_{\Omega} \end{equation}