View source for Standard probability space

''Also: Lebesgue-Rokhlin space''

{{MSC|28Axx|28A50,60A10}}

[[Category:Classical measure theory]]

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A [[probability space]] is called '''standard''' if it satisfies the following equivalent conditions:
*  it is [[Measure space#Isomorphism|almost isomorphic]] to the real line  with some [[probability distribution]] (in other words, a [[Measure  space#Completion|completed]] [[Borel measure|Borel]] [[probability  measure]], that is, a [[Lebesgue–Stieltjes integral|Lebesgue–Stieltjes]]  probability measure);
* it is a [[standard Borel space]] endowed  with a [[probability measure]], completed, and possibly augmented with a  [[Measure space#null|null set]];
* it is [[Measure  space#Completion|complete]], [[Measure space#Perfect and  standard|perfect]], and the [[Hilbert space#L2 space|corresponding  Hilbert space]] is separable.

====The isomorphism theorem====

Every  standard probability space consists of an [[Measure space#Atoms and  continuity|atomic]] (discrete) part and an atomless (continuous) part  (each part may be empty). The discrete part is finite or countable;  here, all subsets are  measurable, and the probability of each subset is  the sum of probabilities of its elements.

'''Theorem 1.''' All atomless standard probability spaces are mutually almost isomorphic.

That   is, up to almost isomorphism we have "the" atomless standard  probability space. Its "incarnations" include the spaces $\R^n$ with  atomless probability distributions ([[Continuous  distribution|absolutely continuous]], [[Singular distribution|singular]]  or mixed), as well as the set of all continuous functions  $[0,\infty)\to\R$ with the [[Wiener measure]]. That is instructive:  topological notions such as dimension do not apply to probability  spaces.

====Measure preserving maps====

The  inverse to a bijective [[Measure space#measure preserving|measure  preserving]] map is measure preserving provided that it is measurable;  in this (not general) case the given map is a [[Measure  space#Isomorphism|strict isomorphism]]. Here is an important fact in two  equivalent forms.

'''Theorem 2a.''' Every bijective measure preserving map between standard probability spaces is a strict isomorphism.

'''Theorem  2b.''' If $(\Om,\F,P)$ is a standard probability space and  $\F_1\subset\F$ a sub-σ-field such that $(\Om,\F_1,P|_{\F_1})$ is also  standard then $\F_1=\F$. 

Recall a topological fact  similar to Theorem 2: if a bijective map  between compact Hausdorff  topological spaces is continuous then it is a homeomorphism. Moreover,  if a Hausdorff topology is  weaker than a compact topology then these  two topologies are equal,  which has the following measure-theory  counterpart stronger than Theorem 2 (in two equivalent forms). 
Here  we call a probability space ''countably separated'' if its underlying  measurable space is [[Measurable space#separated|countably separated]].

'''Theorem  3a.''' Every bijective measure preserving map from a standard  probability space to a  countably separated complete probability space   is a strict isomorphism.

'''Theorem 3b.''' If  $(\Om,\F,P)$ is a standard probability space and $\F_1\subset\F$ is a  countably separated sub-σ-field then $(\Om,\F,P)$ is the completion of  $(\Om,\F_1,P|_{\F_1})$.

A continuous image of a compact  topological space is always a compact set. In contrast, the image of a  measurable set under a (non-bijective) measure-preserving map need not  be measurable (indeed, the image of a null set need not be null; try the  projection $\R^2\to\R^1$). Nevertheless, Theorem 4 (below) is a partial  measure-theory counterpart, stronger than Theorem 3.

'''Theorem  4.''' Let $(\Om,\F,P)$ be a standard probability space,  $(\Om_1,\F_1,P_1)$ a countably separated complete probability space, and  $f:\Om\to\Om_1$ a measure preserving map. Then $(\Om_1,\F_1,P_1)$ is  also standard, and $A_1\in\F_1\iff A\in\F$ whenever $A_1\subset\Om_1$  and $A=f^{-1}(A_1)$. In particular, the image $f(\Om)$belongs to $\F_1$.  (See {{Cite|Ru|Th. 3-2}} and {{Cite|H|Prop. 9}}.)

====Quotient spaces====

Theorem  4 (above) will be combined with the bijective correspondence between  sub-σ-fields, measure subalgebras and linear sublattices described in the [[Measure  space#Sub-σ-algebras, measure subalgebras, and linear sublattices|corresponding section of  "Measure space"]]. The language of linear lattices is used below, but the language of measure subalgebras can be used equally well. Here, as well as there, ''we restrict ourselves to  σ-fields that contain all null  sets.'' 

Every measure  preserving map $\alpha:\Om\to\Om'$ between standard probability spaces  $(\Om,\F,P)$ and $(\Om',\F',P')$ leads to an embedding $f\mapsto  f\circ\alpha$ of Hilbert spaces, $L_2(\Om',\F',P')\to L_2(\Om,\F,P)$. It  is, moreover, an embedding of linear lattices, and therefore  $L_2(\Om',\F',P')=L_2(\Om,\F_1,P|_{\F_1})$ (both embedded into  $L_2(\Om,\F,P)$) for some sub-σ-field $\F_1\subset\F$. Clearly, $\F_1$  is generated by $\alpha$ (up to the null sets), and we may say that  $(\Om',\F',P')$ is the ''quotient space'' of $(\Om,\F,P)$ by $\F_1$ (via  $\alpha$).

''Existence.'' Let $(\Om,\F,P)$ be a  standard probability space and $\F_1\subset\F$ a sub-σ-field; then  $\F_1$ is generated by some $\alpha$ (as above), which means existence  of a quotient space of $(\Om,\F,P)$ by $\F_1$. Here is how to do it.  Using separability of $L_2(\Om,\F_1,P|_{\F_1})$ one constructs a  measurable map $\alpha:\Om\to\Om'$ from $(\Om,\F,P)$ to a standard  measurable space $(\Om',\B)$ such that a function of $L_2(\Om,\F,P)$  belongs to $L_2(\Om,\F_1,P|_{\F_1})$ if and only if it is of the form  $g\circ\alpha$ for some measurable $g:\Om'\to\R$. Taking the image of  the measure $P$ under $\alpha$ and applying Theorem 4 one gets a  standard probability space $(\Om',\F',P')$ and a measure preserving map  $\alpha:\Om\to\Om'$ that generates $\F_1$.

''Uniqueness.''  If also $(\Om'',\F'',P'')$ is the quotient space of $(\Om,\F,P)$ by  $\F_1$ (via $\beta$) then there exists an almost isomorphism $\gamma$  from $(\Om',\F',P')$ to $(\Om'',\F'',P'')$ such that  $\gamma\circ\alpha=\beta$, which means uniqueness of the quotient space  up to almost isomorphism.

Existence of $\gamma$ (above)  follows from the following fact. Let $(\Om,\F,P)$, $(\Om',\F',P')$ and  $(\Om'',\F'',P'')$ be standard probability spaces, and  $\alpha:\Om\to\Om'$, $\beta:\Om\to\Om''$ measure preserving maps. If the  sub-σ-field generated by $\beta$ is contained in the sub-σ-field  generated by $\alpha$ then $\beta=\gamma\circ\alpha$ for some (almost  unique) measure preserving map $\gamma:\Om'\to\Om''$. This is basically  the Doob-Dynkin lemma.

Let $(\Om,\F,P)$ be a standard  probability space, $\F_1,\F_2\subset\F$ two [[Independence#independent  sigma-fields|independent]] sub-σ-fields, and $(\Om',\F',P')$,  $(\Om'',\F'',P'')$ the corresponding quotient spaces (via $\alpha$,  $\beta$); then the product space $(\Om',\F',P')\times(\Om'',\F'',P'')$  is the quotient space of $(\Om,\F,P)$ by $\sigma(\F_1,\F_2)$ (via  $\alpha\times\beta:\omega\mapsto(\alpha(\omega),\beta(\omega))$). Here  $\sigma(\F_1,\F_2)$ is the sub-σ-field generated by $\F_1,\F_2$. If, in  addition, $\sigma(\F_1,\F_2)=\F$ then $\alpha\times\beta$ is an almost  isomorphism from $(\Om,\F,P)$ to $(\Om',\F',P')\times(\Om'',\F'',P'')$.  In this sense, any two independent sub-σ-fields $\F_1,\F_2$ that  generate $\F$ decompose $(\Om,\F,P)$ into the product of two standard  probability spaces (quotient spaces). The same holds for any finite or  countable family of independent sub-σ-fields.

====Conditional measures====

Let $\alpha:\Om\to\Om'$ be a measure preserving map between standard probability spaces $(\Om,\F,P)$ and $(\Om',\F',P')$.

'''Theorem  5a''' ''(existence).'' There exist families  $(\F_{\om'})_{\om'\in\Om'}$, $(P_{\om'})_{\om'\in\Om'}$ of σ-fields  $\F_{\om'}$ on $\Om$ and probability measures $P_{\om'}$ on  $(\Om,\F_{\om'})$ such that for almost every $\om'\in\Om'$
* $(\Om,\F_{\om'},P_{\om'})$ is a standard probability space,
* $\alpha(\om)=\om'$ for $P_{\om'}$-almost all $\om\in\Om$,
and for every $A\in\F$
* the function $\om'\mapsto P_{\om'}(A)$ on $(\Om',\F',P')$ is measurable,
* $P(A)=\int_{\Om'} P_{\om'}(A)\,P'(\!\rd\om')$.

'''Theorem  5b''' ''(uniqueness).'' If also families $(\F'_{\om'})_{\om'\in\Om'}$,  $(P'_{\om'})_{\om'\in\Om'}$ satisfy the requirements of Theorem 5a then  $\F_{\om'}=\F'_{\om'}$ and $P_{\om'}=P'_{\om'}$ for almost all  $\om'\in\Om'$.

The measure $P_{\om'}$ is called the  conditional measure on the subset $\{\om:\alpha(\om)=\om'\}$ of $\Om$,  or the conditional distribution of $\om$ given $\alpha(\om)=\om'$.

'''Example.'''  The projection $(x,y)\mapsto x$ from the square $(0,1)\times(0,1)$ with  the two-dimensional Lebesgue measure to the interval $(0,1)$ with the  one-dimensional Lebesgue measure is a measure preserving map. The  conditional distribution of $(x,y)$ given $x$ is the one-dimensional  Lebesgue measure on the interval $\{x\}\times(0,1)$ with the  one-dimensional Lebesgue measure. This example is trivial, but note the  different σ-fields: neither $\F_{\om'}\subset\F$ nor  $\F\subset\F_{\om'}$.

====On history====

The  notion (but not the term) "standard probability space" and Theorem 1  (isomorphism) were published by Halmos and von Neumann in 1942  {{Cite|HN}}, and by Rokhlin in 1949 {{Cite|Ro}} following Rokhlin's  unpublished manuscript of 1940 (according to {{Cite|Ro|p. 2}}). Theorem 2 is due to  Halmos and von Neumann {{Cite|HN|Sect. 2, Lemma 3 on p. 337}}. Theorems 3 and 4 are due to Rokhlin, see {{Cite|Ro|Sect. 2.5, p. 22}} and {{Cite|Ro|Sect. 3.2}} respectively. Theorem 5  (conditional measures) is due to Rokhlin {{Cite|Ro|Sect. 3.1}}.

====References====

{|
|valign="top"|{{Ref|I}}||   Kiyosi Itô, "Introduction to probability theory", Cambridge (1984).  &nbsp; {{MR|0777504}} &nbsp; {{ZBL|0545.60001}}
|-
|valign="top"|{{Ref|Ru}}||  Thierry de la Rue, "Espaces de Lebesgue", ''Séminaire de Probabilités  XXVII,'' Lecture Notes in Mathematics, '''1557''' (1993), Springer, Berlin,  pp. 15–21. &nbsp;  {{MR|1308547}} &nbsp; {{ZBL|0788.60001}}
|-
|valign="top"|{{Ref|H}}||   Jean Haezendonck, "Abstract  Lebesgue-Rohlin  spaces",  ''Bull. Soc.   Math. de Belgique'' '''25'''   (1973), 243–258.   &nbsp;    {{MR|0335733}} &nbsp;   {{ZBL|0308.60006}}
|-
|valign="top"|{{Ref|HN}}|| P.R. Halmos, J. von Neumann, "Operator  methods in classical mechanics, II", ''Annals of Mathematics (2)''  '''43''' (1942), 332–350.   &nbsp;   {{MR|0006617}} &nbsp;    {{ZBL|0063.01888}}
|-
|valign="top"|{{Ref|Ro}}||  V.A. Rokhlin, (1962), "On the fundamental ideas of measure theory",  ''Translations (American Mathematical Society) Series 1,'' 10 (1962),  1–54. &nbsp; {{MR|0047744}} &nbsp; Translated from Russian:  Рохлин, В. А. (1949), "Об основных понятиях теории меры", Математический  Сборник (Новая Серия) 25(67): 107–150. &nbsp; {{MR|0030584}}
|}