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On page 52 of Noncommutative Geometry (available here: http://www.alainconnes.org/docs/book94bigpdf.pdf), Alain Connes states,

"This wealth of transformations of a measure space $X$ is bound up in the existence, up to isomorphism, of only one interesting measure space: an interval equipped with Lebesgue measure."

What is the more precise version of this thought? (i.e. what does interesting mean?) And where can I find a proof?

I found a relevant theorem on page 279 in volume two of Bogachev's Measure Theory:

9.3.4. Theorem. Every separable atomless measure algebra is isomorphic to the measure algebra of some interval with Lebesgue measure.

The Encyclopedia of Math calls this the isomorphism theorem (https://www.encyclopediaofmath.org/index.php/Measure_algebra_%28measure_theory%29).

hardmath
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Ben
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1 Answers1

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It seems that the following discussion may be of interest. There a formal definition of the standard probability space: Bogachev's result is surely relevant here - however I would not say that all interesting probability spaces are atomless: note that a standard probability space may have atoms. Interesting there most likely means: found in applications, like constructions of stochastic processes etc.

There are several equivalence results in this area, one of the finest one being Borel isomorphism theorem which essentially says that any "interesting" measurable space is isomorphic to a unit interval with the usual $\sigma$-algebra. Note that here we talk about isomorphisms between measurable (not measure) spaces, so that measures are not fixed yet, and thus we can't talk about isomorphism mod 0 so far. Hence, Borel isomoprhism theorem imposes stronger conditions, however even they are usually enough it most non-artificial probabilistic/measure-theoretical constructions.

SBF
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  • So, I think a possible formulation of what Connes is saying is that given a standard Borel space, if we take any partition into standard Borel sub-spaces (precisely the $G_{\delta}$ sets), then since these are classified by their cardinality, any permutation of the equipotent sub-spaces (via any measurable isomorphisms) defines a transformation of the space (an isomorphism of measurable spaces). – Ben Mar 11 '15 at 23:45