# What is the space, on which a random variable is defined?

Well, I have some simple, maybe silly question about random variables, but there is something that I can not undestand when we define them. Suppese that, we have some random variable $$X$$, that is defined in a standard probability space $$(\Omega,\mathcal{F}_s,\mathbb{P})$$, where $$\mathbb{P}:\mathcal{F}_s\rightarrow [0,1]$$. I struggle to undestand which is the space , that this random variable is defined. Specifically, the random variable $$X$$ is a function $$X:\Omega\rightarrow R$$, where R is some arbitrary space and probabyly the real line. Can we claim that $$X\in L^2(\Omega,\mathcal{F}_s,\mathbb{P})$$ or this is wrong? How can we know indeed where this random variable belongs to?

Maybe my whole skeptic is wrong, so forgive me in advance, but I am a begginer, who wants to understand, this mathematical theory!

• It is $X:\Omega\rightarrow\mathbb{R}$. Mar 20, 2020 at 14:52
• @Michael You are right! Mar 20, 2020 at 15:26

Here's a few of the basic definitions.

Given your setup, a random variable is a function $$X : \Omega \to \mathbb R$$, that is, it is a function whose domain is $$\Omega$$.

But not just any old function from $$\Omega$$ to $$\mathbb R$$ is allowed as a random variable. There is a requirement, namely that for every open interval $$(a,b) \subset \mathbb R$$, the set $$X^{-1}(a,b) = \{\omega \in \Omega \mid X(\omega) \in (a,b)\}$$ is an element of the $$\sigma$$-algebra $$\mathcal F_s$$. In other words, $$X^{-1}(a,b)$$ is required to be an event.

Because of this requirement, we can define the expectation of a random variable $$X$$ as an integral, namely $$E(X) = \int_\Omega X \, d \mathbb P$$ Now as I have stated this so far, that integral might be infinite. So, one might wish to impose one more requirement on the definition of a random variable, namely that $$E(X)$$ is finite. This would be equivalent to requiring that $$X \in L^1(\Omega, \mathcal F_s, \mathbb P)$$.

• thank you for your answear, it is very helpful, but I have something in mind that I want to make it clear a little further. Usually, I think that we define random variables $X(\omega,t)$ where we say that if we fix time, then it is a finction of the sample space $\Omega$, we can have in mind a Brownian motion, or if we fix it as function of time $t$, where as we know $t\in [0+\infty)$. Is this completely different from my question in the beginning? Mar 20, 2020 at 15:03
• It looks like you are thinking about a stochastic process, which is a deeper concept that a random variable. But that's another question. Mar 20, 2020 at 15:21
• yes you are right, I just figured out the same think! Well in such a case stochastic process is some $X\in L^2(\Omega,\mathcal{F}_s,\mathbb{P})$ Mar 20, 2020 at 15:23
• so to understand the concept in depth, if a function is a random variable and it has a finiete integral then, we can say that it holds $X\in L^1(\Omega,\mathcal{F}_s,\mathbb{P})$. In addition to this, if the random variable is a square integrable function, then it holds that it also has a finite variance and as a consequence $X\in L^2(\Omega,\mathcal{F}_s,\mathbb{P})$ or $\mathbb{V}ar(X)+\int_{A}(X-\mathbb{E}(X))f(X)dX<\infty$, $A\in\Omega$. Am I right? Mar 23, 2020 at 9:38
• That sounds about right. Mar 23, 2020 at 14:04