Namespaces
Variants
Actions

Difference between revisions of "User:Boris Tsirelson/sandbox1"

From Encyclopedia of Mathematics
Jump to: navigation, search
Line 3: Line 3:
 
\newcommand{\B}{\mathcal B}
 
\newcommand{\B}{\mathcal B}
 
\newcommand{\M}{\mathcal M} $
 
\newcommand{\M}{\mathcal M} $
The term "universally measurable" may be applied to
+
The term '''"universally measurable"''' may be applied to
 
* a [[measurable space]];
 
* a [[measurable space]];
 
* a subset of a measurable space;
 
* a subset of a measurable space;
 
* a [[metric space]].
 
* a [[metric space]].
  
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,
+
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) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad
 
\mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,.$
 
\mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,.$
 
(See {{Cite|S|p. 170}}.)
 
(See {{Cite|S|p. 170}}.)
  
A 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}}.)
+
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}}.)
 +
 
 +
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.
  
 
====References====
 
====References====

Revision as of 07:41, 17 February 2012

$\newcommand{\Om}{\Omega} \newcommand{\A}{\mathcal A} \newcommand{\B}{\mathcal B} \newcommand{\M}{\mathcal M} $ The term "universally measurable" may be applied to

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 [S, p. 170].)

A separable metric space is called universally measurable if it is a universally measurable subset (as defined above) of its completion. Here the completion, endowed with the Borel σ-algebra, is treated as a measurable space. (See [S, p. 170], [D, Sect. 11.5].)

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.

References

[S] Rae M. Shortt, "Universally measurable spaces: an invariance theorem and diverse characterizations", Fundamenta Mathematicae 121 (1984), 169–176.   MR0765332   Zbl 0573.28018
[P] David Pollard, "A user's guide to measure theoretic probability", Cambridge (2002).   MR1873379   Zbl 0992.60001
[K] Alexander S. Kechris, "Classical descriptive set theory", Springer-Verlag (1995).   MR1321597   Zbl 0819.04002
[BK] Howard Becker and Alexander S. Kechris, "The descriptive set theory of Polish group actions", Cambridge (1996).   MR1425877   Zbl 0949.54052
[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
[H] Paul R. Halmos, "Measure theory", v. Nostrand (1950).   MR0033869   Zbl 0040.16802
[R] Walter Rudin, "Principles of mathematical analysis", McGraw-Hill (1953).   MR0055409   Zbl 0052.05301
How to Cite This Entry:
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=21110