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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 [[Lebesgue–Stieltjes integral|
 
Lebesgue–Stieltjes measure]];
 
* it is a [[standard Borel space]] endowed with a [[probability measure]], [[Measure space#Completion|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 a set 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.
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{|
 
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| A || B || C
====Measure preserving injections====
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|-
 
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| X || Y || Z
Here is another important fact in two equivalent forms.
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|}
 
 
'''Theorem 2a.''' If a bijective map between standard probability spaces is measure preserving then the inverse map is also measure preserving.
 
 
 
'''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 standard then $\F_1=\F$.
 
 
 
-------------------------------------------------------
 
 
 
(See Definition 1 below.) Every standard probability space is isomorphic (mod 0) to an interval with [[Lebesgue measure]], a finite or countable set of atoms, or a combination of both. (See Theorem ? below.)
 
 
 
''Example.'' The set of all continuous functions $[0,\infty)\to\R$ with the [[Wiener measure]] is a standard probability space.
 
  
''Non-example.'' The set $[0,1]^\R$ of all functions $\R\to[0,1]$ with the product of Lebesgue measures is a nonstandard probability space.
 
  
'''Definition 1a.''' A probability space $(\Om,\F,P)$ is ''standard'' if it is [[Measure space#complete|complete]] and there exist a subset $\Om_1\subset\Om$ and a σ-field (in other words, σ-algebra) $\B$ on $\Om_1$ such that $(\Om_1,\B)$ is a standard Borel space and every set of $\F$ is [[Measure space#almost|almost equal]] to a set of $\B$. (See {{Cite|I|Sect. 2.4}}.) (Clearly, $\Om_1$ must be of [[Measure space#full|full measure]].)
 
  
'''Definition 1b''' (equivalent). A probability space $(\Om,\F,P)$ is ''standard'' if it is complete, [[Measure space#perfect|perfect]] and countably separated mod 0 in the following sense: some subset of full measure, treated as a [[Measurable space#subspace|subspace]] of the measurable space $(\Om,\F)$, is a [[Measurable space#separated|countably separated]] measurable space.
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-----------------------------------------
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-----------------------------------------
  
(See {{Cite|I|Sect. 3.1}} for a proof of equivalence of these definitions.)
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$\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$
  
====On terminology====
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<asy>
 +
size(100,100);
 +
label(scale(1.7)*'$T(\\Sigma)\hookrightarrow T(\\Sigma,X)$',(0,0));
 +
</asy>
  
Also "Lebesgue-Rokhlin space" and "[[Lebesgue space]]".
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<asy>
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size(220,220);
  
In {{Cite|M|Sect. 6}} universally measurable spaces are called metrically standard Borel spaces.
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import math;
  
In {{Cite|K|Sect. 21.D}} universally measurable subsets of a standard (rather than arbitrary) measurable space are defined.
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int kmax=40;
  
In {{Cite|N|Sect. 1.1}} an absolute measurable space is defined as a separable metrizable topological space such that every its homeomorphic image in every such space (with the Borel σ-algebra) is a universally measurable subset. The corresponding measurable space (with the Borel σ-algebra) is also called an absolute measurable space in {{Cite|N|Sect. B.2}}.
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guide g;
 +
for (int k=-kmax; k<=kmax; ++k) {
 +
  real phi = 0.2*k*pi;
 +
  real rho = 1;
 +
  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);
  
====References====
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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 );
  
{|
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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|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


A B C
X Y Z




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

How to Cite This Entry:
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=21360