Definition

measurable map

If $(\Omega, \mathcal{B})$ and $(\Omega ', \mathcal{B} ')$ are two measurable spaces, then a map $X: \Omega \to \Omega '$ is called measurable if $X^{-1}(\mathcal{B} ') \subset \mathcal{B}$. In this case we write $X \in \mathcal{B} / \mathcal{B}'$ or $X: (\Omega, \mathcal{B}) \to (\Omega ', \mathcal{B} ')$. If $(\Omega ', \mathcal{B} ') = (\mathbb{R}, \mathcal{B}(\mathbb{R}))$ then we call $X$ a random variable.