Difference between revisions of "User:Boris Tsirelson/sandbox1"
From Encyclopedia of Mathematics
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\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 | + | 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 | + | 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
- a measurable space;
- a subset of a measurable 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,
- $ \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
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=21110