Difference between revisions of "User:Boris Tsirelson/sandbox1"
From Encyclopedia of Mathematics
| Line 1: | Line 1: | ||
| + | $\newcommand{\Om}{\Omega} | ||
| + | \newcommand{\A}{\mathcal A} | ||
| + | \newcommand{\B}{\mathcal B} | ||
| + | \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, | ||
| + | : $ \mu_*(A) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad | ||
| + | \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
- 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\}\,.$
How to Cite This Entry:
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=21099
Boris Tsirelson/sandbox1. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox1&oldid=21099