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Having put my audience through developing a probability space $(\Omega, \mathcal{F}, P)$, where $\Omega$ is the sample space of a random trial, $\mathcal{F}$ is a $\sigma$-field on it, and P is a measure on that $\sigma$-field, how to I motivate shifting the whole construct to the real line?

And while I'm at it, how do I justify it informally, i.e., without heavy math? I know (I think, but I'm a bit fuzzy) that a random variable $X:\Omega \rightarrow \mathbb{R}$ induces a Borel field but even if I'm right about that how does that convince the listener that the whole probability space is ported to the real line?

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    $\begingroup$ Is this a course for mathematicians or for people who apply probability theory? $\endgroup$ Commented Mar 30, 2022 at 7:29
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    $\begingroup$ General answer: Get a good textbook and follow it. There are many levels of probability courses. Some do not talk about "probability space" at all. $\endgroup$ Commented Mar 30, 2022 at 12:26
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    $\begingroup$ What do you want to motivate and justify, exactly? The definition of a random variable $X \colon \Omega \to \mathbb{R}$, the notion of its distribution on $\mathbb{R}$, the notion of $X$-measurable events, or something else? In any case, I don't see heavy math involved except for the things you already take for granted - i.e. being familiar with measures and measurability. $\endgroup$ Commented Mar 30, 2022 at 13:04
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    $\begingroup$ I agree with @MichaelBächtold, there is no answer for this question if the course audience is not specified. $\endgroup$
    – Opal E
    Commented Mar 30, 2022 at 15:22
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    $\begingroup$ I imagine the best answer will be a combination of @GeraldEdgar 's comment and a specific textbook. $\endgroup$ Commented Mar 30, 2022 at 19:12

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