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\newcommand{\M}{\mathcal M} $
 
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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.
 
''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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(See {{Cite|S|p. 170}}, {{Cite|D|Sect. 11.5}}.)
 
 
 
 
 
 
 
 
The term '''"universally measurable"''' may be applied to
 
* a [[measurable space]];
 
* a subset of a measurable space;
 
* a [[metric space]].
 
 
 
'''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,
 
: $ \mu_*(A) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad
 
\mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,.$
 
(See {{Cite|C|Sect. 8.4}}, {{Cite|S|p. 170}}.)
 
 
 
Universally measurable sets evidently are a σ-algebra that contains the σ-algebra $\A$ of measurable sets.
 
 
 
''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.
 
 
 
'''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}}.)
 
  
 
'''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}}.)
 
'''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.
 
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.
  
'''Theorem 1''' (Shortt {{Cite|S|Theorem 1}}). The following two conditions on a separable metric space are equivalent:
 
:(a) it is a universally measurable metric space;
 
:(b) the corresponding measurable space (with the Borel σ-algebra) is universally measurable.
 
 
Evidently, (a) implies (b); surprisingly, also (b) implies (a), which  involves a Borel isomorphism (rather than isometry or homeomorphism)  between two metric spaces.
 
 
'''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]].
 
 
Every standard Borel space evidently is universally  measurable. And moreover:
 
  
'''Theorem 3.''' Every [[analytic Borel space]] is universally  measurable.
 
  
 
====On terminology====
 
====On terminology====

Revision as of 09:28, 18 February 2012

$\newcommand{\Om}{\Omega} \newcommand{\F}{\mathcal F} \newcommand{\B}{\mathcal B} \newcommand{\M}{\mathcal M} $ 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.)

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 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 $\forall A\in\F \;\; \exists B\in\B \; \big( B \subset A \land P(B)=P(A) \big)$.


(See [S, p. 170], [D, Sect. 11.5].)

Definition 3. A measurable space is called universally measurable if it is isomorphic to some universally measurable metric space (as defined above) with the Borel σ-algebra. (See [S, p. 171].)

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.


On terminology

In [M, Sect. 6] universally measurable spaces are called metrically standard Borel spaces.

In [K, Sect. 21.D] universally measurable subsets of a standard (rather than arbitrary) measurable space are defined.

In [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 [N, Sect. B.2].

References

[I] Kiyosi Itô, "Introduction to probability theory", Cambridge (1984).   MR0777504   Zbl 0545.60001
[C] Donald L. Cohn, "Measure theory", Birkhäuser (1993).   MR1454121   Zbl 0860.28001
[D] Richard M. Dudley, "Real analysis and probability", Wadsworth&Brooks/Cole (1989).   MR0982264   Zbl 0686.60001
[M] George W. Mackey, "Borel structure in groups and their duals", Trans. Amer. Math. Soc. 85 (1957), 134–165.   MR0089999   Zbl 0082.11201
[K] Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).   MR1321597   Zbl 0819.04002
[N] Togo Nishiura, "Absolute measurable spaces", Cambridge (2008).   MR2426721   Zbl 1151.54001
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Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=21167