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I have a simple question about the meaning of random variables in probability. I understand how one can define a probability space $(\Omega, \Sigma, P)$ and then a (real) random variable $X : \Omega \to \mathbb{R}$ which, if continuous, defines a PDF $f(x)$ such that

$$P(a \leq X \leq b) = \int_a^b f(x) dx .$$

This direction is fine, but in practice I think the reasoning is in the opposite direction and I don't see how. For instance say you have a Gaussian PDF

$$f(x) = \frac{1}{\sqrt{2 \pi}} e^{-x^2/2} .$$

I think it is common to say that this defines a gaussian random variable, but what exactly is this random variable? What is the probability space and what is the function $X: \Omega \to \mathbb{R}$?

One possible answer (see e.g. this post) is that the PDF defines a probability space with $\Omega=\mathbb{R}$ and $P([a,b]) = \int_a^b f(x) dx$, so that $f(x)$ is the PDF of the "identity" random variable $X : \mathbb{R} \to \mathbb{R}$. But then how should I understand more complex situations like convergence in distribution $X_n \to X$? I think $X$ and the $X_n$'s should be defined on the same probability space. So what is this space?

MBolin
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    Kolmogorov's consistency Theorem (aka Kolmogorov's existence theoerm) is a very general result about construction of random variables. – geetha290krm Jun 29 '23 at 11:27
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    For convergence in distribution you don't need the random variables to be defined on the same space. This type of convergence depends only on the CDF. In general, most of the time we care only about the random variables and not on the particular space where they are defined. We know such a space exists. (even more so, given any collection of distributions, there exists a space where we can define infinitely many independent random variables with such distributions) – Mark Jun 29 '23 at 11:29
  • Ok you don't need them in the same probability space. I guess that solves solves my question. Also good to know about Kolmogorov's existence thm. – MBolin Jun 29 '23 at 11:37

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No, a probability measure $\mu$ in $\mathbb{R}$ doesn't define a specific random variable, as it doesn't defines a specific function $X:\Omega \to \mathbb{R}$. What it defines, given a probability space $(\Omega ,\mathcal{S}, P)$, is the family of functions $\{X\in \mathcal{L}_0(P): P\circ X^{-1}=\mu\}$.

Note: above $\mathcal{L}_0(P)$ is the set of measurable functions $X:\Omega \to \mathbb{R}$.

Masacroso
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