Borel space
Definition 0.1
A
Borel space

is defined as a set

, together with
a Borel
-algebra

of subsets of

, called
Borel sets. The Borel algebra on

is the smallest

-algebra containing all open sets (or, equivalently, all closed sets if the topology on closed sets is selected).
Remark 0.1
Borel sets were named after the French mathematician Emile Borel.
Remark 0.2
A subspace of a Borel space

is a subset

endowed with the relative Borel structure, that is the

-algebra of all subsets of

of the form

, where

is a Borel subset of

.
Definition 0.2
A
rigid Borel space

is defined as a Borel space whose only automorphism

(that is, with

being a bijection, and also with

for any

) is the
identity
function

(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'.
-
- 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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