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''Also: Lebesgue-Rokhlin space''
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<ref> [http://hea-www.harvard.edu/AstroStat http://hea-www.harvard.edu/AstroStat]; <nowiki> http://www.incagroup.org </nowiki>; <nowiki> http://astrostatistics.psu.edu </nowiki> </ref>
  
{{MSC.|28Axx|28A50,60A10}}
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====Notes====
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<references />
  
[[:Category:Classical measure theory]]
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-------------------------------------------
  
{{TEX|done}}
 
  
$\newcommand{\Om}{\Omega}
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{|
\newcommand{\om}{\omega}
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| A || B || C
\newcommand{\F}{\mathcal F}
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|-
\newcommand{\B}{\mathcal B}
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| X || Y || Z
\newcommand{\M}{\mathcal M} $
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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 (be they [[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|R|Th. 3-2}} and {{Cite|H|Prop. 9}}.)
 
 
 
====Quotient spaces====
 
 
 
Theorem 4 (above) will be combined with the bijective correspondence between sub-σ-fields and linear sublattices described in the [[Measure space#Sub-σ-algebras and linear sublattices|corresponding section of "Measure space"]]. 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.
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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.
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$\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$
  
====Conditional measures====
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<asy>
 +
size(100,100);
 +
label(scale(1.7)*'$T(\\Sigma)\hookrightarrow T(\\Sigma,X)$',(0,0));
 +
</asy>
  
Let $\alpha:\Om\to\Om'$ be a measure preserving map between standard probability spaces $(\Om,\F,P)$ and $(\Om',\F',P')$.
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<asy>
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size(220,220);
  
'''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 $\om'\in\Om'$
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import math;
* $(\Om,\F_{\om'},P_{\om'})$ 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 $\om'\in\Om'$.
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int kmax=40;
  
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 $\alpha(\om)=\om'$.
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guide g;
 +
for (int k=-kmax; k<=kmax; ++k) {
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  real phi = 0.2*k*pi;
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  real rho = 1;
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  if (k!=0) {
 +
    rho = sin(phi)/phi;
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  }
 +
  pair z=rho*expi(phi);
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  g=g..z;
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}
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draw (g);
  
'''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 $\F\subset\F_{\om'}$.
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defaultpen(0.75);
 +
draw ( (0,0)--(1.3,0), dotted, Arrow(SimpleHead,5) );
 +
dot ( (1,0) );
 +
label ( "$a$", (1,0), NE );
  
====References====
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</asy>
 
 
{|
 
|valign="top"|{{Ref|I}}||  Kiyosi Itô, "Introduction to probability theory", Cambridge (1984). &nbsp; {{MR|0777504}} &nbsp; {{ZBL|0545.60001}}
 
|-
 
|valign="top"|{{Ref|R}}||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}}
 
|}
 

Latest revision as of 07:12, 13 March 2016

[1]

Notes

  1. http://hea-www.harvard.edu/AstroStat; http://www.incagroup.org ; http://astrostatistics.psu.edu


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$\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$

How to Cite This Entry:
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=21478