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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{\A}{\mathcal A}
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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 the definition below.)
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| | | | |
| | + | ====Notes==== |
| | + | <references /> |
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| | + | ------------------------------------------- |
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| | | | |
| | + | {| |
| | + | | A || B || C |
| | + | |- |
| | + | | X || Y || Z |
| | + | |} |
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| | | | |
| | + | ----------------------------------------- |
| | + | ----------------------------------------- |
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| − | The term '''"universally measurable"''' may be applied to
| + | $\newcommand*{\longhookrightarrow}{\lhook\joinrel\relbar\joinrel\rightarrow}$ |
| − | * a [[measurable space]];
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| − | * a subset of a measurable space;
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| − | * a [[metric space]].
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| − | | |
| − | '''Definition 1.''' Let $(X,\A)$ be a measurable space. A ''subset'' $A\subset X$ is called ''universally measurable'' if it is $\mu$-measurable for every finite measure $\mu$ on $(X,\A)$. In other words: $\mu_*(A)=\mu^*(A)$ where $\mu_*,\mu^*$ are the inner and outer measures for $\mu$, that is,
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| − | : $ \mu_*(A) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad
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| − | \mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,.$
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| − | (See {{Cite|C|Sect. 8.4}}, {{Cite|S|p. 170}}.)
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| − | | |
| − | Universally measurable sets evidently are a σ-algebra that contains the σ-algebra $\A$ of measurable sets.
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| − | | |
| − | ''Warning.'' Every measurable set is universally measurable, but an universally measurable set is generally not measurable! This terminological anomaly appears because the word "measurable" is used differently in two contexts, of measurable spaces and of measure spaces.
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| − | | |
| − | '''Definition 2.''' A separable ''metric space'' is called ''universally measurable'' if it is a universally measurable subset (as defined above) of its [[Metric space#completion|completion]]. Here the completion, endowed with the [[Measurable space#Borel sets|Borel σ-algebra]], is treated as a measurable space. (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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| − | | |
| − | '''Theorem 1''' (Shortt {{Cite|S|Theorem 1}}). The following two conditions on a separable metric space are equivalent:
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| − | :(a) it is a universally measurable metric space;
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| − | :(b) the corresponding measurable space (with the Borel σ-algebra) is universally measurable.
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| − | | |
| − | Evidently, (a) implies (b); surprisingly, also (b) implies (a), which involves a Borel isomorphism (rather than isometry or homeomorphism) between two metric spaces.
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| − | '''Theorem 2''' (Shortt {{Cite|S|Lemma 4}}). A [[Measurable space#countably generated|countably generated]] [[Measurable space#separated|separated]] measurable space $(X,\A)$ is universally measurable if and only if for every finite measure $\mu$ on $(X,\A)$ there exists a subset $A\in\A$ of full measure (that is, $\mu(X\setminus A)=0$) such that $A$ (treated as a [[Measurable space#subspace|subspace]]) is itself a [[standard Borel space]].
| + | <asy> |
| | + | size(100,100); |
| | + | label(scale(1.7)*'$T(\\Sigma)\hookrightarrow T(\\Sigma,X)$',(0,0)); |
| | + | </asy> |
| | | | |
| − | Every standard Borel space evidently is universally measurable. And moreover:
| + | <asy> |
| | + | size(220,220); |
| | | | |
| − | '''Theorem 3.''' Every [[analytic Borel space]] is universally measurable.
| + | import math; |
| | | | |
| − | ====On terminology==== | + | int kmax=40; |
| | | | |
| − | In {{Cite|M|Sect. 6}} universally measurable spaces are called metrically standard Borel spaces.
| + | 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); |
| | | | |
| − | In {{Cite|K|Sect. 21.D}} universally measurable subsets of a standard (rather than arbitrary) measurable space are defined.
| + | defaultpen(0.75); |
| | + | draw ( (0,0)--(1.3,0), dotted, Arrow(SimpleHead,5) ); |
| | + | dot ( (1,0) ); |
| | + | label ( "$a$", (1,0), NE ); |
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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}}.
| + | </asy> |
| − | | |
| − | ====References====
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| − | | |
| − | {|
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| − | |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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