In mathematics, a Borel isomorphism is a measurable bijective function between two measurable
standard Borel spaces. By
Souslin's theorem in standard Borel spaces (a set that is both
analytic and
coanalytic is necessarily Borel), the inverse of any such measurable bijective function is also measurable. Borel isomorphisms are closed under composition and under taking of inverses. The set of Borel isomorphisms from a space to itself clearly forms a
group
A group is a number of persons or things that are located, gathered, or classed together.
Groups of people
* Cultural group, a group whose members share the same cultural identity
* Ethnic group, a group whose members share the same ethnic iden ...
under composition. Borel isomorphisms on standard Borel spaces are analogous to
homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphi ...
s on
topological space
In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called points ...
s: both are bijective and closed under composition, and a homeomorphism and its inverse are both
continuous
Continuity or continuous may refer to:
Mathematics
* Continuity (mathematics), the opposing concept to discreteness; common examples include
** Continuous probability distribution or random variable in probability and statistics
** Continuous ...
, instead of both being only Borel measurable.
Borel space
A
measurable space
In mathematics, a measurable space or Borel space is a basic object in measure theory. It consists of a set and a σ-algebra, which defines the subsets that will be measured.
Definition
Consider a set X and a σ-algebra \mathcal A on X. Then the ...
that is Borel isomorphic to a measurable subset of the real numbers is called a Borel space.
See also
*
Federer–Morse theorem
In mathematics, the Federer–Morse theorem, introduced by , states that if ''f'' is a surjective continuous map from a compact metric space ''X'' to a compact metric space ''Y'', then there is a Borel subset ''Z'' of ''X'' such that ''f'' re ...
References
*
Alexander S. Kechris (1995) ''Classical Descriptive Set Theory'', Springer-Verlag.
[ {{cite book , last1=Kallenberg , first1=Olav , author-link1=Olav Kallenberg , year=2017 , title=Random Measures, Theory and Applications, location= Switzerland , publisher=Springer , page=15, doi= 10.1007/978-3-319-41598-7, isbn=978-3-319-41596-3 ]
External links
* S. K. Berberian (1988
Borel Spacesfrom
University of Texas
The University of Texas at Austin (UT Austin, UT, or Texas) is a public research university in Austin, Texas. It was founded in 1883 and is the oldest institution in the University of Texas System. With 40,916 undergraduate students, 11,075 ...
*
Richard M. Dudley (2002
Real Analysis and Probability, 2nd edition page 487.
* Sashi Mohan Srivastava (1998
A Course on Borel Sets
Measure theory