View source for Analytic Borel space

''Also: analytic measurable space''

[[Category:Descriptive set theory]]
[[Category:Classical measure theory]]

{{User:Rehmann/sandbox/MSC|03E15|28A05,54H05}}

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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.

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]).

''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].)

====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].)

As every subset of a measurable space, an analytic set is itself a measurable space (a 
[[Measurable space#subspace|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 [[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.

====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]).

'''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 [[Measurable space#separating|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.

====References====
{|
|valign="top"|[1]||  Alexander S.  Kechris, "Classical descriptive set theory",  Springer-Verlag (1995). &nbsp; {{MR|1321597}}  &nbsp;{{ZBL|0819.04002}} 
|-
|valign="top"|[2]|| George  W. Mackey,  "Borel structure in groups and their duals", ''Trans.  Amer.  Math. Soc.''  '''85''' (1957), 134–165. &nbsp; {{MR|0089999}}  &nbsp;   {{ZBL|0082.11201}}
|-
|valign="top"|[3]|| S.M.  Srivastava, "A course on Borel sets", Springer-Verlag (1998). &nbsp;  {{MR|1619545}} &nbsp;{{ZBL|0903.28001}} 
|}