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| − | $\newcommand{\Om}{\Omega}
| + | <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}
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| − | \newcommand{\B}{\mathcal B}
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| − | \newcommand{\M}{\mathcal M} $
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| − | A [[probability space]] is called '''standard''' if it satisfies the following equivalent conditions:
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| − | * 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);
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| − | * it is a [[standard Borel space]] endowed with a [[probability measure]], completed, and possibly augmented with a [[Measure space#null|null set]];
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| − | * it is [[Measure space#Completion|complete]], [[Measure space#Perfect and standard|perfect]], and the [[Hilbert space#L2 space|corresponding Hilbert space]] is separable.
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| − | ====The isomorphism theorem==== | + | ====Notes==== |
| | + | <references /> |
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| − | 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.
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| − | 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.
| + | {| |
| − | | + | | A || B || C |
| − | ====Measure preserving maps====
| + | |- |
| − | | + | | X || Y || Z |
| − | 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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| − | | |
| − | '''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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| − | 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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| − | Here we call a probability space ''countably separated'' if its underlying measurable space is [[Measurable space#separated|countably separated]].
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| − | | |
| − | '''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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| − | | |
| − | '''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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| − | 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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| − | | |
| − | '''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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| − | -------------------------------------------------------
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| − | ''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.
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| − | '''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]].)
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| − | '''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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| − | (See {{Cite|I|Sect. 3.1}} for a proof of equivalence of these definitions.)
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| − | ====Quotient spaces====
| + | ----------------------------------------- |
| | + | ----------------------------------------- |
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| − | 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 $\F_1\subset\F$.
| + | $\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$ |
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| − | ====On terminology====
| + | <asy> |
| | + | size(100,100); |
| | + | label(scale(1.7)*'$T(\\Sigma)\hookrightarrow T(\\Sigma,X)$',(0,0)); |
| | + | </asy> |
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| − | Also "Lebesgue-Rokhlin space" and "[[Lebesgue space]]".
| + | <asy> |
| | + | size(220,220); |
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| − | In {{Cite|M|Sect. 6}} universally measurable spaces are called metrically standard Borel spaces.
| + | import math; |
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| − | In {{Cite|K|Sect. 21.D}} universally measurable subsets of a standard (rather than arbitrary) measurable space are defined.
| + | int kmax=40; |
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| − | 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}}.
| + | guide g; |
| | + | for (int k=-kmax; k<=kmax; ++k) { |
| | + | real phi = 0.2*k*pi; |
| | + | real rho = 1; |
| | + | if (k!=0) { |
| | + | rho = sin(phi)/phi; |
| | + | } |
| | + | pair z=rho*expi(phi); |
| | + | g=g..z; |
| | + | } |
| | + | |
| | + | draw (g); |
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| − | ====References====
| + | defaultpen(0.75); |
| | + | draw ( (0,0)--(1.3,0), dotted, Arrow(SimpleHead,5) ); |
| | + | dot ( (1,0) ); |
| | + | label ( "$a$", (1,0), NE ); |
| | | | |
| − | {|
| + | </asy> |
| − | |valign="top"|{{Ref|I}}|| Kiyosi Itô, "Introduction to probability theory", Cambridge (1984). {{MR|0777504}} {{ZBL|0545.60001}}
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| − | |-
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| − | |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. {{MR|1308547}} {{ZBL|0788.60001}}
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| − | |-
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| − | |valign="top"|{{Ref|H}}|| Jean Haezendonck, "Abstract Lebesgue-Rohlin spaces", ''Bull. Soc. Math. de Belgique'' '''25''' (1973), 243–258. {{MR|0335733}} {{ZBL|0308.60006}}
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| − | |-
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| − | |valign="top"|{{Ref|B}}|| V.I. Bogachev, "Measure theory", Springer-Verlag (2007). {{MR|2267655}} {{ZBL|1120.28001}}
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| − | |-
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| − | |valign="top"|{{Ref|C}}|| Donald L. Cohn, "Measure theory", Birkhäuser (1993). {{MR|1454121}} {{ZBL|0860.28001}}
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| − | |-
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| − | |valign="top"|{{Ref|D}}|| Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989). {{MR|0982264}} {{ZBL|0686.60001}}
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| − | |-
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| − | |valign="top"|{{Ref|M}}|| George W. Mackey, "Borel structure in groups and their duals", ''Trans. Amer. Math. Soc.'' '''85''' (1957), 134–165. {{MR|0089999}} {{ZBL|0082.11201}}
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| − | |-
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| − | |valign="top"|{{Ref|K}}|| Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995). {{MR|1321597}} {{ZBL|0819.04002}}
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| − | |-
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| − | |valign="top"|{{Ref|N}}|| Togo Nishiura, "Absolute measurable spaces", Cambridge (2008). {{MR|2426721}} {{ZBL|1151.54001}}
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| − | |}
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