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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.

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$X^{-1}(B)$ is the set of all states of some probabilistic system such that the parameters determined by $X$ have values lying in $B$. You want to assign a probability to this happening, so it needs to be measurable. In other words, you want to be able to define

$\mathbb{P}(X \in B) = \mu(X^{-1}(B)).$

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    Th$a$t fact should have made it obvious. Thanks for clarifying!2012-10-02