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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 is a [[standard Borel space]] endowed with a probability measure, completed with null sets, and possibly augmented with another null set. (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.)
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| − | ''Example.'' The set of all continuous functions $[0,\infty)\to\R$ with the Wiener measure is a standard probability space.
| + | ====Notes==== |
| | + | <references /> |
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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 1.''' A probability space $(\Om,\F,P)$ is ''standard'' if it is complete (that is, $\F$ contains all null sets; these are sets $A\subset\Om$ such that $ \exists B\in\F \big( A\subset B \land P(B)=0 \big) $) 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 $\forall A\in\F \;\; \exists B\in\B \; \big( B \subset A \land P(B)=P(A) \big)$.
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| | + | {| |
| | + | | A || B || C |
| | + | |- |
| | + | | X || Y || Z |
| | + | |} |
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| − | (See {{Cite|S|p. 170}}, {{Cite|D|Sect. 11.5}}.)
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| − | '''Definition 3.''' A ''measurable space'' is called ''universally measurable'' if it is [[Measurable space#isomorphic|isomorphic]] to some universally measurable metric space (as defined above) with the Borel σ-algebra. (See {{Cite|S|p. 171}}.)
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| − | Thus, the phrase "universally measurable space" is ambiguous; it can be interpreted as "universally measurable metric space" or "universally measurable measurable space"! The latter can be replaced with "universally measurable Borel space", but the ambiguity persists. Fortunately, the ambiguity is rather harmless by the following result.
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| | + | $\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$ |
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| | + | <asy> |
| | + | size(100,100); |
| | + | label(scale(1.7)*'$T(\\Sigma)\hookrightarrow T(\\Sigma,X)$',(0,0)); |
| | + | </asy> |
| | | | |
| − | ====On terminology====
| + | <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|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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