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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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I believe these isomorphic spaces are called "standard." See this Wiki link for example: en.wikipedia.org/wiki/Standard_probability_space – Alex R. Jul 7 '12 at 9:18
    
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. – JL344 Jul 7 '12 at 22:37

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