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Finite and countable analytic Borel spaces are trivial: all subsets are measurable. Uncountable  
 
Finite and countable analytic Borel spaces are trivial: all subsets are measurable. Uncountable  
analytic Borel spaces are of cardinality continuum. Some, but not all, of them are standard; these are mutually isomorphic. Some additional (to ZFC) set-theoretic axioms imply that all nonstandard analytic Borel spaces are mutually isomorphic (see [1, Sect. 26.D]).
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analytic Borel spaces are of cardinality continuum. Some, but not all, of them are standard; these are mutually isomorphic. Some additional (to [[ZFC]]) set-theoretic axioms imply that all nonstandard analytic Borel spaces are mutually isomorphic (see [1, Sect. 26.D]).
  
 
====References====
 
====References====

Revision as of 12:16, 25 January 2012

Also: analytic measurable space

Category:Classical measure theory

[ 2010 Mathematics Subject Classification MSN: 28A05,(03E15,54H05) | MSCwiki: 28A05   + 03E15,54H05  ]

$ \newcommand{\R}{\mathbb R} \newcommand{\C}{\mathbb C} \newcommand{\Om}{\Omega} \newcommand{\A}{\mathcal A} \newcommand{\B}{\mathcal B} \newcommand{\P}{\mathbf P} $ A Borel space $(X,\A)$ is called analytic if it is countably separated and isomorphic to a quotient space of a standard Borel space.

This is one out of several equivalent definitions (see below).

Finite and countable analytic Borel spaces are trivial: all subsets are measurable. Uncountable analytic Borel spaces are of cardinality continuum. Some, but not all, of them are standard; these are mutually isomorphic. Some additional (to ZFC) set-theoretic axioms imply that all nonstandard analytic Borel spaces are mutually isomorphic (see [1, Sect. 26.D]).

References

[1] Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).   MR1321597  Zbl 0819.04002
[2] George W. Mackey, "Borel structure in groups and their duals", Trans. Amer. Math. Soc. 85 (1957), 134–165.   MR0089999   Zbl 0082.11201
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Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=20473