Disappearance of sample space

In probability theory, the "orthodox approach" (Kolmogorov approach?) defines events, probability measure, random variable, etc., in terms of a sample space (often denoted $\Omega$). However, after a certain point, the sample space "disappears" or fades into the background.

At a certain point in most probability courses, the sample space is rarely mentioned anymore and we work directly with random variables. But you should keep in mind that the sample space is really there, lurking in the background.

Warning! We defined random variables to be mappings from a sample space $\Omega$ to $\mathbb {R}$ but we did not mention the sample space in any of the distributions above. As I mentioned earlier, the sample space often "disappears" but it is really there in the background. Let's construct a sample space explicitly for a Bernoulli random variable. Let $\Omega =[0,1]$ and define $\mathbb {P}$ to satisfy $\mathbb {P} ([a,b])=b-a$ for $0\leq a\leq b\leq 1$. Fix $p\in [0,1]$ and define

$$X(\omega )={\begin{cases}1&\omega \leq p\\0&\omega >p.\end{cases}}$$

Then $\mathbb {P} (X=1)=\mathbb {P} (\omega \leq p)=\mathbb {P} ([0,p])=p$ and $P(X=0)=1-p$. Thus, $X\sim \mathrm {Bernoulli} (p)$. We could do this for all the distributions defined above. In practice, we think of a random variable like a random number but formally it is a mapping defined on some sample space.

Now that we understand probability triples well, we discuss some additional essential ingredients of probability theory. Throughout Section 3 (and, indeed, throughout most of this text and most of probability theory in general), we shall assume that there is an underlying probability triple $(\Omega ,{\mathcal {F}},\mathbf {P} )$ with respect to which all further probability objects are defined. This assumption shall be so universal that we will often not even mention it.

In Klenke's book there is a theorem, 1.104, "For any distribution function F, there exists a real random variable X with F_X = F."

See also tao's two blog posts. [1] [2]

So in the end i think there are two ways to think about this:

• you can always start with the random variable and go back to make up some (silly/artificial/mickey mouse) sample space that works, so let's just forget about sample spaces
• we can think of our sample space as continually expanding to accommodate our ever-growing demand of sources of randomness; but thankfully, probability theory only studies concepts preserved under expanding our sample space so we can do this.
• is there a third way to think about this, in terms of world histories?

I think we also want a theorem like this: if $X\colon \Omega \to R$ and $Y\colon \Omega '\to S$ are two random variables in different probability spaces, then we can get some extension over a sample space $\Omega ''$ that somehow combines $\Omega ,\Omega '$ (maybe a simple product would work?). Actually maybe this is underspecified somehow, e.g. would X and Y be independent? (this question sort of doesn't make sense since X and Y were separated to begin with)

See also

• Summary table of probability terms
• List of sample space perspectives

External links

• https://math.stackexchange.com/questions/23006/the-role-of-the-hidden-probability-space-on-which-random-variables-are-defined?rq=1
• https://math.stackexchange.com/questions/1612012/how-should-i-understand-the-probability-space-omega-mathcalf-p-what-d
• https://math.stackexchange.com/questions/2531810/why-does-probability-theory-insist-on-sample-spaces?rq=1
• https://math.stackexchange.com/questions/712734/domain-of-a-random-variable-sample-space-or-probability-space?rq=1
• https://math.stackexchange.com/questions/18198/what-are-the-sample-spaces-when-talking-about-continuous-random-variables/18199#18199