Probability – What are the Sample Spaces for Continuous Random Variables?

Question

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?

Answer 1

You can take it to be a subset of $\mathbb{R}$ or, more generally, $\mathbb{R}^n$. A random variable uniformly distributed in $[0, 1]$ can be thought of as a random variable on the sample space $[0, 1]$ with probability density function $1$.

The sample space is in fact not critically important. (You may find it convenient to pick one to do computations in, but it doesn't matter which one you pick. This is analogous to choosing coordinates to do computations in linear algebra.) This point is explained very clearly in Terence Tao's notes here:

At a purely formal level, one could call probability theory the study of measure spaces with total measure one, but that would be like calling number theory the study of strings of digits which terminate. At a practical level, the opposite is true [emphasis added]: just as number theorists study concepts (e.g. primality) that have the same meaning in every numeral system that models the natural numbers, we shall see that probability theorists study concepts (e.g. independence) that have the same meaning in every measure space that models a family of events or random variables. And indeed, just as the natural numbers can be defined abstractly without reference to any numeral system (e.g. by the Peano axioms), core concepts of probability theory, such as random variables, can also be defined abstractly, without explicit mention of a measure space; we will return to this point when we discuss free probability later in this course.

Terence Tao's explanation of free probability is here. (I found it very enlightening; the framework Tao describes can be used to describe quantum probability with very little modification, unlike the measure theory framework.)