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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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I guess I'm asking an even more elementary question. How is "$\mathbf{X}^{−1}(B)$ the set of all states of some probabilistic system such that the parameters determined by $\mathbf{X}$ have values lying in $\mathbf{B}$"? –  Luis Costa Oct 2 '12 at 1:35
    
@Luis: $\Omega$ is the set of states, $X$ is a collection of $n$ parameters $X_1, ... X_n : \Omega \to \mathbb{R}$, and $X^{-1}(B)$ is by definition the set of all $s \in \Omega$ such that $(X_1(s), ... X_n(s)) \in B$. Think of very simple examples, for example take $\Omega$ to be the set of all possible outcomes of two dice rolls, $X_1$ to be the outcome of the first roll, and $X_2$ to be the outcome of the second, $B$ to be the event $X_1 + X_2 = 7$ or anything you like. –  Qiaochu Yuan Oct 2 '12 at 1:44
    
Are you saying that the inverse notation $X^{-1}$ is just a convention in probability theory and not really an inverse mapping of some sort? I'm reading this like the typical inverse of a function with the property $f^{-1}(f(x))=x$. –  Luis Costa Oct 2 '12 at 1:51
    
No. $X^{-1}$ denotes the inverse image (en.wikipedia.org/wiki/Image_(mathematics)#Inverse_image). –  Qiaochu Yuan Oct 2 '12 at 2:02
    
Ah. From the wiki "The two coincide only if $f$ is a bijection". I'm guessing they do not in this case? This is frustrating. It seems no reference I have makes that distinction. I am a graduate student in theoretical and computational mechanics so I must have missed some of the pre-requisite math to make this distinction. I knew the differences between the two in general (from elementary analysis). Where would I have picked up this distinction just based on the definition I started with? –  Luis Costa Oct 2 '12 at 2:13

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