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