| Version current |
Version 15 |
| \begin{definition} |
\begin{definition} |
| A {\em Borel space} $(X; \mathcal{B}(X))$ is defined as a set $X$, together with |
A {\em Borel space} $(X; \mathcal{B}(X))$ is defined as a set $X$, together with |
| a \PMlinkname{$\sigma$-algebra}{SigmaAlgebra} $\mathcal{B}(X)$ of subsets of $X$, called Borel sets. |
a \PMlinkname{$\sigma$-algebra}{SigmaAlgebra} $\mathcal{B}(X)$ of subsets of $X$, called Borel sets. |
| \end{definition} |
\end{definition} |
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| \begin{remark} |
\begin{remark} |
| 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 $X$. |
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 $X$. |
| \end{remark} |
\end{remark} |
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| \begin{definition} |
\begin{definition} |
| A \emph{rigid Borel space} $(X_r; \mathcal{B} (X_r))$ is defined as a Borel space whose only automorphism |
A \emph{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.\cite{BA91}). |
$f: X_r \to X_r$ (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.\cite{BA91}). |
| \end{definition} |
\end{definition} |
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| \begin{remark} |
\begin{remark} |
| R. M. Shortt and J. Van Mill provided the first construction of a rigid Borel space on a `set of large cardinality'. |
R. M. Shortt and J. Van Mill provided the first construction of a rigid Borel space on a `set of large cardinality' |
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(that is, on a class). |
| \end{remark} |
\end{remark} |
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| \begin{thebibliography}{9} |
\begin{thebibliography}{9} |
|
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| \bibitem{MRB2k6} |
\bibitem{MRB2k6} |
| M.R. Buneci. 2006., |
M.R. Buneci. 2006., |
| \PMlinkexternal{Groupoid C*-Algebras.}{http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf}, |
\PMlinkexternal{Groupoid C*-Algebras.}{http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf}, |
| {\em Surveys in Mathematics and its Applications}, Volume 1: 71--98. |
{\em Surveys in Mathematics and its Applications}, Volume 1: 71--98. |
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| \bibitem{BA91} |
\bibitem{BA91} |
| B. Aniszczyk. 1991. A rigid Borel space., {\em Proceed. AMS.}, 113 (4):1013-1015., |
B. Aniszczyk. 1991. A rigid Borel space., {\em Proceed. AMS.}, 113 (4):1013-1015., |
| \PMlinkexternal{available online}{http://www.jstor.org/pss/2048777}. |
\PMlinkexternal{available online}{http://www.jstor.org/pss/2048777}. |
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| \bibitem{AC79} |
\bibitem{AC79} |
| A. Connes.1979. Sur la th\'eorie noncommutative de l' integration, {\em Lecture Notes in |
A. Connes.1979. Sur la th\'eorie noncommutative de l' integration, {\em Lecture Notes in |
| Math.}, Springer-Verlag, Berlin, {\bf 725}: 19-14. |
Math.}, Springer-Verlag, Berlin, {\bf 725}: 19-14. |
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| \end{thebibliography} |
\end{thebibliography} |
