Why can distributions be completely defined as probability measures on $\mathbb{R}$? In my textbook, it states that 
"Part of the beauty of distributions comes from the fact that two statisticians, who are in different parts of the world working on completely different problems with completely different probability spaces, may both find the same distribution extremely useful; they can even discuss this distribution together in the common language of probability measures on $\mathbb{R}$, without worrying about whether they have completely different $\Omega$'s."
I am a bit confused by this statement because I don't get how one can have different $\Omega$'s without changing the problem at hand. Is there a trivial way to see why this is true?
 A: It means that there is no difference in observable outcomes whether you flip a coin or roll a die with three heads and three tails - two different $\Omega$'s, same distribution of outcomes.
ETA: For example, while you can construct $X\sim B(1,1/2)$ on a single coin, you can't construct a pair of independent identically distributed $X,Y$ random variables on the same underlying $\Omega$ - you need to expand your underlying $\Omega$ to two coins or one four-sided die. The whole "probability space" shebang is a mathematical machinery to rigorously justify something experienced naturally (for example what it means for observations to be independent). That's why you have theorems showing that there actually exist probability spaces and random variables that have given distributions.
After we hide the probability space, we you no longer speak of convergence a.s. or convergence in probability - only of convergence in distribution. This is useful because in physical reality we don't really have access to the actual source of randomness ($\Omega$) - only to observables/outcomes and their (empirical) distributions.
