Reading the answer of this question (which is very good), https://stats.stackexchange.com/questions/141416/example-of-sample-x-1-x-2-ldots-x-n/141574#comment846751_141574, I realised that there is a point I don't understand. I quote bullet point two:
- such an experiment can be, e.g.: pick a random person in Paris and measure them. The output is a size. Each particular experiment leads a size $x$, which is a real number. $\underline{\textit{We denote $X$ the random variable which is associated }}$; $X$ is not a number. A way to see it is as function from the set $\Omega$ of all possible experiment results, in our case the set of Parisians, to the real numbers. In this optic, a random variable is a measure done on a random experiment.
The underlined sentence is what I don't understand. How this random variable $X : \Omega \to \mathbb{R}$ is defined? In other words, given $x \in \mathbb{R}$, that is interpreted as a height of a Parisian, how the corresponding random variable $X$ is be defined (or, constructed)? I'm not necessarily talking about explicit formula (if a "natural" map exists of course will be happy to see it), but the rationale behind this identification.
As a sidenote, I'm aware of the realisation of a random variable and I understand that given $x \in \mathbb{R}$, for the associated random variable we mean that $X(\omega)=x$, for some $\omega \in \Omega$. What I don't understand is how $X(\omega^{\prime})$ should be understood, for $\omega^{\prime} \neq \omega$.