Probabilists don't care, what exactly the domain of random variables is. Here is an extreme comment that exemplifies this: "You should soon see, if you learn more stochastic stuff, that specifying the underlying probability space is a BAD IDEA (what happens when you add a new head/tails?), and quite useless.
If specifying the underlying probability space domain of a random variable (in short "domain" from now on) is such a useless, bad idea for most scenarios, I'm wondering why no one in the long history of probability theory resp. statistics has come up with a better, slicker definition of random variables, that avoids this unelegant we-have-a-domain-but-we-won't-talk-about-it situation?
It seems the only reason to keep the domain $\Omega$ is to enabling a coupling of random variables, so that we can speak of their independence. But can't such a coupling be realized in a more elegant way, than using a space that we don't want to define in the first place?
As soon as I'm reading texts that go beyond very elementary probability, it seems to me that such domains are treated like the crazy uncle from family parties: which we never show them/him, but know it's there.
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