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I stumbled upon this exercise in my Probability Theory professor's notes:

Determine whether you can create a probability space $(\Omega, \mathcal{F}, P)$ with an infinite amount of random variables $X_1, X_2, ..., X_n,...$ so that they are all independent and they all have the same distribution. If so, state your example.

I don't really understand it to be honest. I believe such a space can indeed be built but I wouldn't know why, like for example, an infinite amount of independent Poisson r.v.s but how can I determine $\Omega, \mathcal{F}$? I'd have to set an experiment in particular and the possible results right?

I'm just taking this class but this sort of questions really interest me.

MNM
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    Any point of $(0,1)$ has an expansion $x=\sum \frac {x_n} {2^{n}}$ , $x_i \in {0,1}$. With Lebesgue measure on this set you get i.i.d. random variable $x \to x_i, i=1,2...$. – Kavi Rama Murthy May 07 '21 at 10:21
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    Here is another example. –  May 07 '21 at 10:23
  • Could you guys please elaborate on the examples? In the first example $\Omega = (0,1)$, how are the random variables exactly defined? as $x$? What about $\mathcal{F}$? – MNM May 07 '21 at 14:35
  • @M.Navarro: There are good explanations about "canonical" probability spaces upon which one can develop the theory of probability (see Kallenberg, Foundations of Probability, for instance, to Rick Durret, Probability.) Here I gave it a try to explain this things. This include's Kavi Rama Murthy's example. The random variables in his example are the bits $x_n(\omega)$ of $\omega\in[0,1)$ in its binary expansion. – Mittens May 08 '21 at 15:25

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