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After looking for a dozen of questions on cross validated, I've decided to write my own. We know the following mapping is called a random variable $$X:\Omega\to \mathbb{R}$$ where $\Omega$ is set of all possible outcomes. For simple stochastic experiments like coin tossing we know the sample space explicitly is $\Omega= \text{\{Head, Tail\}}$. However, in most cases when we use random variables and we only know the distribution or not even that; we certainly ignore the sample space let alone its interpretation. I am concerned about the interpretations of some "real life" random variables. As an example let $H$ be the distribution representing the height of the person. assume $X \sim H$. Let me also define the experiment: picking a person randomly and getting his/her height.

Question

How can we interpret $\Omega = \{\omega_1,\omega_2, \omega_3,...\}$

each $\omega_i$ in my intuition cannot be simply the all possible 7 billion people on earth, I guess it must all possible 'types' of people, it is like a latent feature that describes a particular type of a possible person in some sense? Therefore, it takes into account even people how are not even born? Or is it possible to have a discrete $\Omega$ for this problem whatsoever? And if yes what is that mysterious $w_i$.

P.S. If possible, also tell whether H must have any hypothetical parameters.

If there is some clear why to explain this mystery, I beg your help.

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    $\begingroup$ In standard usage, the meanings of the arrows $\to,\mapsto$ are different. For that reason I changed $X:\Omega\mapsto\mathbb R$ to $X:\Omega\to\mathbb R. \qquad$ $\endgroup$ – Michael Hardy Feb 4 at 20:50
  • $\begingroup$ @MichaelHardy what's the difference? $\endgroup$ – Erik Hambardzumyan Feb 4 at 20:51
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    $\begingroup$ The second arrow is used in things like $x\mapsto x^3,$ which means the function whose output is the cube of its input. $\endgroup$ – Michael Hardy Feb 4 at 20:51
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    $\begingroup$ The question is using $H$ with two different meanings. $\endgroup$ – Xi'an Feb 5 at 6:06
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    $\begingroup$ @user158565 I'm having a hard time seeing how measure theory has anything at all to say about interpreting statistical concepts. $\endgroup$ – whuber Feb 7 at 22:16
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Suppose $\Omega$ is some population of persons and $\omega\in\Omega,$ i.e. $\omega$ is one of those persons. Further suppose each person has some probability of being the one who is chosen. Then $X(\omega)$ is a random variable whose distribution depends on the probabilities of various persons being chosen and also on the values of $X$ at the various values of $\omega.$

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  • $\begingroup$ edited the distribution thing, was a typo $\endgroup$ – Erik Hambardzumyan Feb 4 at 21:02
  • $\begingroup$ so $w_i$ is the person? $\endgroup$ – Erik Hambardzumyan Feb 4 at 21:03
  • $\begingroup$ The question was is $\Omega$ technically an infinite set $\endgroup$ – Erik Hambardzumyan Feb 4 at 21:04
  • $\begingroup$ The answer is not satisfying please give a detailed explanation $\endgroup$ – Erik Hambardzumyan Feb 9 at 11:52
  • $\begingroup$ by your reasoning $\omega_i$ is one of 7 billion people existing which is not technically true $\endgroup$ – Erik Hambardzumyan Feb 9 at 12:00

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