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单词 BorelSpace
释义

Borel space


Definition 0.1.

A Borel spacePlanetmathPlanetmath (X;(X)) is defined as a set X, together witha Borel σ-algebra (http://planetmath.org/SigmaAlgebra) (X) of subsets of X, called Borel sets. The Borel algebra on X is the smallest σ-algebra containing all open sets (or, equivalently, all closed setsPlanetmathPlanetmath if the topologyMathworldPlanetmath on closed sets is selected).

Remark 0.1.

Borel sets were named after the French mathematician Emile Borel.

Remark 0.2.

A subspaceMathworldPlanetmathPlanetmath of a Borel space (X;(X)) is a subset SX endowed with the relative Borel structure, that is the σ-algebra of all subsets of S of the form SE, where E is a Borel subset of X.

Definition 0.2.

A rigid Borel space (Xr;(Xr)) is defined as a Borel space whose only automorphismMathworldPlanetmathPlanetmathPlanetmathPlanetmathf:XrXr (that is, with f being a bijectionMathworldPlanetmath, and also with f(A)=f-1(A) for any A(Xr)) is the identity function 1(Xr;(Xr)) (ref.[2]).

Remark 0.3.

R. M. Shortt and J. Van Mill provided the first construction of a rigid Borel space on a ‘set of large cardinality’.

References

  • 1 M.R. Buneci. 2006.,http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdfGroupoidPlanetmathPlanetmathPlanetmathPlanetmath C*-Algebras.,Surveys in Mathematics and its Applications, Volume 1: 71–98.
  • 2 B. Aniszczyk. 1991. A rigid Borel space., Proceed. AMS., 113 (4):1013-1015.,http://www.jstor.org/pss/2048777available online.
  • 3 A. Connes.1979. Sur la théorie noncommutative de l’ integration, Lecture Notes inMath., Springer-Verlag, Berlin, 725: 19-14.
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