TITLE: Basic formality when considering random numbers
QUESTION [6 upvotes]: Suppose we are interested in randomly picking numbers in the interval $[0,1]$ with the uniform distribution. If I want to write a mathematical text about this, it can be done by saying that $X$ is a random variable with uniform distribution in $[0,1]$, or $X\sim U[0,1]$ for short. Then I can analyze the expected value of $X$, it's variance, some particular probabilities and so on. By analyzing $X$, I'm analyzing the process of picking numbers described above. 
In practice, I'm able to do that, and if asked to formalize more about this random variable, I would introduce a probability space $(\Omega, \mathcal{F}, P)$, say that $X:\Omega\to[0,1]$ is a measurable function, talk about it's density function and so on. Everything ok to this point. 
Usually the sample space $\Omega$ stands for some concrete (real world) problem we are interested in and $X$ translates this concrete in numbers (so we can use mathematics). But in this case there is no concrete, we are only interested in numbers. We really just want the codomain of $X$, not the domain. So I considered two explanations of how we should interpret $\Omega$:
1) We just let $\Omega$ undefined. It is there just to formalize the notion of a measurable function. 
2) $\Omega = [0,1]$ and $X(\omega) = \omega$, for all $\omega\in [0,1]$. 
My two questions are: How should we interpret $\Omega$ in this situation? And what are the differences between my two interpretations (is one of them more right)?
Thanks!!!

REPLY [2 votes]: sorry for answering so late!
In all practical terms you would get away with the first option (not specifying what $\Omega$ is). Even academic papers in pure mathematics often omit this, because as you say "we really just want the codomain of $X$".
However if you want to formalise your random variable, I would definitely go with the second option you offered, that is taking $\Omega = [0,1]$ and $X(\omega) = \omega$ for all $\omega\in [0,1]$. The sample space $\Omega$ stands for the set of all possible outcomes/results of some experiment. Here we take the experiment to be picking numbers in the interval $[0,1]$, which may not seem so "real-world" but does not need to be! If you would prefer a more concrete experiment, picking such a number could be interpreted as drawing a unit interval on the wall and blindly throwing a dart on it.
Then the results of the experiment are real numbers from $0$ to $1$, and it becomes clear that we should take $\Omega$ and $X$ as you suggested! :)