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$\newcommand{\Om}{\Omega}
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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>
\newcommand{\F}{\mathcal F}
 
\newcommand{\B}{\mathcal B}
 
\newcommand{\M}{\mathcal M} $
 
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====
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====Notes====
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<references />
  
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.
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-------------------------------------------
  
'''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, connectedness, compactness etc. do not apply to probability spaces.
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{|
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| A || B || C
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|-
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| X || Y || Z
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|}
  
====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.
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-----------------------------------------
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-----------------------------------------
  
'''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 $\F_1=\F$.
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$\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$
  
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).
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<asy>
Here we call a probability space ''countably separated'' if its underlying measurable space is [[Measurable space#separated|countably separated]].
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size(100,100);
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label(scale(1.7)*'$T(\\Sigma)\hookrightarrow T(\\Sigma,X)$',(0,0));
 +
</asy>
  
'''Theorem 3a.''' Every bijective measure preserving map from a standard probability space to a  countably separated complete probability space  is a strict isomorphism.
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<asy>
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size(220,220);
  
'''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 $(\Om,\F_1,P|_{\F_1})$.
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import math;
  
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.
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int kmax=40;
  
'''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}}.)
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guide g;
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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) {
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    rho = sin(phi)/phi;
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  }
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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);
  
====Quotient spaces====
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defaultpen(0.75);
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draw ( (0,0)--(1.3,0), dotted, Arrow(SimpleHead,5) );
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dot ( (1,0) );
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label ( "$a$", (1,0), NE );
  
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 (see [[Measure space#Sub-σ-algebras and linear sublattices]]) 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-algebra $\F_1\subset\F$.
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</asy>
 
 
====On terminology====
 
 
 
Also "Lebesgue-Rokhlin space" and "[[Lebesgue space]]".
 
 
 
====References====
 
 
 
{|
 
|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}}
 
|-
 
|valign="top"|{{Ref|B}}|| V.I. Bogachev, "Measure theory",  Springer-Verlag (2007). &nbsp;  {{MR|2267655}}  &nbsp;{{ZBL|1120.28001}}
 
|-
 
|valign="top"|{{Ref|C}}|| Donald L. Cohn, "Measure theory", Birkhäuser (1993). &nbsp;    {{MR|1454121}} &nbsp;  {{ZBL|0860.28001}}
 
|-
 
|valign="top"|{{Ref|D}}||  Richard M. Dudley, "Real analysis and probability",  Wadsworth&Brooks/Cole (1989). &nbsp; {{MR|0982264}} &nbsp;  {{ZBL|0686.60001}}
 
|-
 
|valign="top"|{{Ref|M}}||  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"|{{Ref|K}}|| Alexander  S.  Kechris,  "Classical  descriptive set theory", Springer-Verlag  (1995).  &nbsp;  {{MR|1321597}} &nbsp; {{ZBL|0819.04002}}
 
|-
 
|valign="top"|{{Ref|N}}|| Togo Nishiura, "Absolute  measurable spaces",  Cambridge (2008). &nbsp;  {{MR|2426721}}  &nbsp;  {{ZBL|1151.54001}}
 
|}
 

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=21441