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A  [[measurable space|Borel space]] is called '''analytic''' if it is [[Measurable space#separated|countably separated]] and [[Measurable  space#isomorphic|isomorphic]] to a [[Measurable space#quotient  space|quotient space]] of a [[Standard Borel space|standard]] Borel  space.
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A  [[measurable space|Borel space]] is called '''analytic''' if it is [[Measurable space#separated|countably separated]] and [[Measurable  space#isomorphic|isomorphic]] to a [[Measurable space#quotient  space|quotient space]] of a [[Standard Borel space|standard]] Borel  space.
 
  
 
See below for an equivalent definition.
 
See below for an equivalent definition.
  
 
Finite and countable analytic Borel spaces are trivial: all subsets are measurable. Uncountable  
 
Finite and countable analytic Borel spaces are trivial: all subsets are measurable. Uncountable  
analytic  Borel spaces are of [[Continuum, cardinality of the|cardinality  continuum]]. Some, but not all, of them are standard; these are mutually  isomorphic. Some additional (to [[ZFC]]) set-theoretic axioms imply  that all nonstandard analytic Borel spaces are mutually isomorphic (see  [1, Sect. 26.D]).
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analytic  Borel spaces are of [[Continuum, cardinality of the|cardinality  continuum]]. Some, but not all, of them are standard; these are mutually  isomorphic. Some additional (to [[ZFC]]) set-theoretic axioms imply  that all nonstandard analytic Borel spaces are mutually isomorphic (see  {{Cite|K|Sect. 26.D}}).
  
''Non-example.'' The [[quotient  group]] $\R/\Q$ (real numbers modulo rational numbers, additive) may be  thought of as a quotient measurable space, $\R$ being endowed with its  Borel σ-algebra. Then $\R/\Q$ is a quotient space of a standard Borel  space, but not an analytic Borel space, because it is not countably  separated. (See [2, Sect. 5].)
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''Non-example.'' The [[quotient  group]] $\R/\Q$ (real numbers modulo rational numbers, additive) may be  thought of as a quotient measurable space, $\R$ being endowed with its  Borel σ-algebra. Then $\R/\Q$ is a quotient space of a standard Borel  space, but not an analytic Borel space, because it is not countably  separated. (See {{Cite|M|Sect. 5}}.)
  
 
====Relations to analytic sets====
 
====Relations to analytic sets====
  
A  subset of a standard Borel (or just countably separated measurable)  space is called ''analytic'' if it is the image of a standard Borel  space under a Borel map. (See [1, Sect. 14.A].)
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A  subset of a standard Borel (or just countably separated measurable)  space is called ''analytic'' if it is the image of a standard Borel  space under a Borel map. (See {{Cite|K|Sect. 14.A}}.)
  
 
As every subset of a measurable space, an analytic set is itself a measurable space (a  
 
As every subset of a measurable space, an analytic set is itself a measurable space (a  
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Equivalence  of the two definitions follows from the [[Standard Borel  space#Blackwell-Mackey theorem|Blackwell-Mackey theorem]] and the  following simple fact: every countably separated measurable space admits  a one-to-one measurable map to a standard Borel space.
 
Equivalence  of the two definitions follows from the [[Standard Borel  space#Blackwell-Mackey theorem|Blackwell-Mackey theorem]] and the  following simple fact: every countably separated measurable space admits  a one-to-one measurable map to a standard Borel space.
  
See [1, Sect. 25.A], [2, Sect. 4] for these, and some other, definitions of analytic sets and spaces.
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See {{Cite|K|Sect. 25.A}}, {{Cite|M|Sect. 4}} for these, and some other, definitions of analytic sets and spaces.
  
 
====Measurable injections====
 
====Measurable injections====
  
Several  [[Standard Borel space#Measurable injections|results on standard Borel  spaces]] generalize to analytic Borel spaces (see [2, Sect. 4], [3, Sect. 4.5]).
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Several  [[Standard Borel space#Measurable injections|results on standard Borel  spaces]] generalize to analytic Borel spaces (see {{Cite|M|Sect. 4}}, {{Cite|S|Sect. 4.5}}).
  
 
'''Theorem 1a.''' If a bijective map between analytic Borel spaces is measurable then the inverse map is also measurable.
 
'''Theorem 1a.''' If a bijective map between analytic Borel spaces is measurable then the inverse map is also measurable.
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The Blackwell-Mackey theorem generalizes readily  from standard to analytic spaces, since a quotient space of an analytic  space evidently is also a quotient space of a standard space.
 
The Blackwell-Mackey theorem generalizes readily  from standard to analytic spaces, since a quotient space of an analytic  space evidently is also a quotient space of a standard space.
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Every analytic Borel space is [[universally measurable]].
  
 
====References====
 
====References====
 
{|
 
{|
|valign="top"|[1]||  Alexander S.  Kechris, "Classical descriptive set theory",  Springer-Verlag (1995).   {{MR|1321597}}   {{ZBL|0819.04002}}  
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|valign="top"|{{Ref|K}}||  Alexander S.  Kechris, "Classical descriptive set theory",  Springer-Verlag (1995).   {{MR|1321597}}   {{ZBL|0819.04002}}  
 
|-
 
|-
|valign="top"|[2]|| George  W. Mackey,  "Borel structure in groups and their duals", ''Trans.  Amer.  Math. Soc.''  '''85''' (1957), 134–165.   {{MR|0089999}}     {{ZBL|0082.11201}}
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|valign="top"|{{Ref|M}}|| George  W. Mackey,  "Borel structure in groups and their duals", ''Trans.  Amer.  Math. Soc.''  '''85''' (1957), 134–165.   {{MR|0089999}}     {{ZBL|0082.11201}}
 
|-
 
|-
|valign="top"|[3]|| S.M.  Srivastava, "A course on Borel sets", Springer-Verlag (1998).    {{MR|1619545}}  {{ZBL|0903.28001}}  
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|valign="top"|{{Ref|S}}|| S.M.  Srivastava, "A course on Borel sets", Springer-Verlag (1998).    {{MR|1619545}}  {{ZBL|0903.28001}}  
 
|}
 
|}

Latest revision as of 20:31, 18 February 2012

Also: analytic measurable space

2020 Mathematics Subject Classification: Primary: 03E15 Secondary: 28A0554H05 [MSN][ZBL]

$\newcommand{\A}{\mathcal A} \newcommand{\B}{\mathcal B}$ A Borel space is called analytic if it is countably separated and isomorphic to a quotient space of a standard Borel space.

See below for an equivalent definition.

Finite and countable analytic Borel spaces are trivial: all subsets are measurable. Uncountable analytic Borel spaces are of cardinality continuum. Some, but not all, of them are standard; these are mutually isomorphic. Some additional (to ZFC) set-theoretic axioms imply that all nonstandard analytic Borel spaces are mutually isomorphic (see [K, Sect. 26.D]).

Non-example. The quotient group $\R/\Q$ (real numbers modulo rational numbers, additive) may be thought of as a quotient measurable space, $\R$ being endowed with its Borel σ-algebra. Then $\R/\Q$ is a quotient space of a standard Borel space, but not an analytic Borel space, because it is not countably separated. (See [M, Sect. 5].)

Relations to analytic sets

A subset of a standard Borel (or just countably separated measurable) space is called analytic if it is the image of a standard Borel space under a Borel map. (See [K, Sect. 14.A].)

As every subset of a measurable space, an analytic set is itself a measurable space (a subspace of the given space).

Definition 2 (equivalent). A Borel space is called analytic if it is isomorphic to an analytic set.

Equivalence of the two definitions follows from the Blackwell-Mackey theorem and the following simple fact: every countably separated measurable space admits a one-to-one measurable map to a standard Borel space.

See [K, Sect. 25.A], [M, Sect. 4] for these, and some other, definitions of analytic sets and spaces.

Measurable injections

Several results on standard Borel spaces generalize to analytic Borel spaces (see [M, Sect. 4], [S, Sect. 4.5]).

Theorem 1a. If a bijective map between analytic Borel spaces is measurable then the inverse map is also measurable.

Theorem 1b. If σ-algebras $\A$, $\B$ on $X$ are such that $\A\subset\B$ and $(X,\A)$, $(X,\B)$ are analytic then $\A=\B$.

Example. The real line with the Lebesgue σ-algebra is not analytic (by Theorem 1b).

Theorem 2a. If a bijective map from an analytic Borel space to a countably separated measurable space is measurable then the inverse map is also measurable.

Theorem 2b. If σ-algebras $\A$, $\B$ on $X$ are such that $\A\subset\B$, $(X,\A)$ is countably separated and $(X,\B)$ is analytic then $\A=\B$.

Theorem 2c. If $(X,\A)$ is an analytic Borel space then $\A$ is generated by every at most countable separating subset of $\A$.

The Blackwell-Mackey theorem generalizes readily from standard to analytic spaces, since a quotient space of an analytic space evidently is also a quotient space of a standard space.

Every analytic Borel space is universally measurable.

References

[K] Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).   MR1321597  Zbl 0819.04002
[M] George W. Mackey, "Borel structure in groups and their duals", Trans. Amer. Math. Soc. 85 (1957), 134–165.   MR0089999   Zbl 0082.11201
[S] S.M. Srivastava, "A course on Borel sets", Springer-Verlag (1998).   MR1619545  Zbl 0903.28001
How to Cite This Entry:
Analytic Borel space. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Analytic_Borel_space&oldid=20783