In probability theory we have this definition:
DEFINITION: Let $(\Omega, \mathcal{U}, P)$ be a probability space. A mapping $\mathbf{X}: \Omega \to \mathbb{R}^n$ is called an n-dimensional random variable if for each $B \in \mathcal{B}$, we have $\mathbf{X}^{-1}(B) \in \mathcal{U}$.
where $\Omega$ is a probability space, $\mathcal{U}$ the $\sigma$-algebra of subsets of $\Omega$, B an event $\in \Omega$, and $\mathcal{B}$ the Borel subsets of $\mathbb{R}^n$.
Can someone explain why defining it as such, with the inverse $\mathbf{X}^{-1}(B) \in \mathcal{U}$, is useful? This seems to be a standard property of Borel measurable mappings, but can someone give an explanation of how it applies to probability? Thank you.