What are the sample spaces when talking about continuous random variables? When talking about continuous random variables, with a particular probability distribution, what are the underlying sample spaces? 
Additionally, why these sample spaces are omitted oftentimes, and one simply says r.v. $X$ follows a uniform distribution on the interval $[0,1]$? Isn't the sample space critically important?

the sample space are the numbers that your random variable X can take. If you have a uniform distribution on the interval (0,1) then the value that your r.v. can take is any thing from O to 1. If your the r.v is outside this interval then your pdf is zero. So that means that the universal space are the all the real numbers but for this sample we are only consern with what happens from [0,1]
