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

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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]].
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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\}\,.$

Revision as of 19:30, 16 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\}\,.$
How to Cite This Entry:
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=21101