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Remark 0.1 A subspace of a Borel space $(X; \mathcal{B} (X))$ is a subset $S \subset X$ endowed with the relative Borel structure, that is the $\sigma$ -algebra of all subsets of $S$ of the form $S \bigcap E$ , where $E$ is a Borel subset of $X$ .
Definition 0.2 A rigid Borel space $(X_r; \mathcal{B} (X_r))$ is defined as a Borel space whose only automorphism $f: X_r \to X_r$ (that is, with $f$ being a bijection, and also with $f(A) = f^{-1}(A)$ for any $A \in \mathcal{B}(X_r)$ ) is the identity function $1_{(X_r; \mathcal{B}(X_r))}$ (ref.[ 2]).
Remark 0.2 R. M. Shortt and J. Van Mill provided the first construction of a rigid Borel space on a `set of large cardinality'.
- 1
- M.R. Buneci. 2006., Groupoid 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., available online.
- 3
- A. Connes.1979. Sur la théorie noncommutative de l' integration, Lecture Notes in Math., Springer-Verlag, Berlin, 725: 19-14.
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See Also: Borel set,  -algebra, measurable space, Borel measure, Borel groupoid, Borel morphism, category of Borel spaces, Borel G-space
| Other names: |
measurable space |
| Also defines: |
rigid Borel space, Borel subset space |
| Keywords: |
Borel space, Borel set,  -algebra, rigid Borel space |
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Cross-references: cardinality, identity function, bijection, automorphism, Borel subset, Borel structure, subspace, Borel sets, subsets
There are 49 references to this entry.
This is version 16 of Borel space, born on 2008-09-15, modified 2009-01-24.
Object id is 11026, canonical name is BorelSpace.
Accessed 2856 times total.
Classification:
| AMS MSC: | 28A05 (Measure and integration :: Classical measure theory :: Classes of sets , measurable sets, Suslin sets, analytic sets) | | | 54H05 (General topology :: Connections with other structures, applications :: Descriptive set theory ) | | | 28A12 (Measure and integration :: Classical measure theory :: Contents, measures, outer measures, capacities) | | | 28C15 (Measure and integration :: Set functions and measures on spaces with additional structure :: Set functions and measures on topological spaces ) | | | 60A10 (Probability theory and stochastic processes :: Foundations of probability theory :: Probabilistic measure theory) |
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Pending Errata and Addenda
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