Somewhere I've heard that Kolmogorov proved that for all practical purposes, the probability space $$(\Omega,\mathcal F,\mathbb P)$$ that he invented could be taken without loss of generality to be the unit interval endowed with the Lebesgue measure, $$([0,1],\mathcal L,\mu),$$ although the mappings necessary to define random variables on such a space are in general highly contrived and hence not very constructive or intuitive. Does anyone have a reference or a pointer where to find this proof or a translation of it?

Somewhat related: The role of the "hidden" probability space on which random variables are defined

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    $\begingroup$ I believe these isomorphic spaces are called "standard." See this Wiki link for example: en.wikipedia.org/wiki/Standard_probability_space $\endgroup$
    – Alex R.
    Jul 7, 2012 at 9:18
  • $\begingroup$ That's right! Now I remember! Thanks for reminding me. That was ages ago and I must have forgotten. It's been 14 years since I took probability. $\endgroup$
    – JL344
    Jul 7, 2012 at 22:37


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