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Let's consider a probability space $(\Omega, \mathcal{F}, P)$ corresponding to experiments on throwing a dice and defined in the following way: $\Omega = \{1, 2, 3, 4, 5, 6\}$, $\mathcal{F} = \{\Omega, \varnothing, \{1, 3, 5\}, \{2, 4, 6\}\}$. So, in this $\sigma$-algebra we only have 4 events: something happened, nothing happened, odd number rolled, even number rolled. Can anyone give me an example of random real variable defined for this probability space?

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    A minor remark: the events $\Omega$ and $\varnothing$ do not describe the fact that *something happened* vs *nothing happened*. Rather they correspond to *something certain happened* vs *something impossible happened* (like, *the result of the throw of the dice is an integer* vs *the result of the throw of the die is $7$*). But *something* always *happens*.2011-06-27

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Yes, $X(\omega)=0$ if $\omega \in \{1,3,5\}$, and $X(\omega)=1$ if $\omega \in \{2,4,6\}$.

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    Or equivalently, since $\Omega$ is finite, for any $x \in \mathbb{R}$ fixed, $\{\omega:X(\omega) = x\} \in \mathcal{F}$.2011-06-19
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The random variables $X$ on this probability space are exactly the functions $X:\Omega\to\mathbb{R}$ such that there exists $a$ and $b$ with $X(\omega)=a$ if $\omega\in\{1,3,5\}$ and $X(\omega)=b$ otherwise. That is, every such $X$ is a random variable and every random variable $X$ is like that.

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    Or in other words, $X$ has to be ${\mathcal F}$-measurable which means it can't depend on any other information than whether the result was even or odd.2011-06-19