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Difference between revisions of "User:Boris Tsirelson/sandbox2"

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A '''measure space''' is a triple $(X,\A,\mu)$ where $X$ is a set, $\A$ a σ-algebra of its subsets, and $\mu:\A\to[0,+\infty]$ a measure.
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A '''measure space''' is a triple $(X,\A,\mu)$ where $X$ is a set, $\A$ a [[Algebra of sets|σ-algebra]] of its subsets, and $\mu:\A\to[0,+\infty]$ a [[measure]]. Thus, a measure space consists of a [[measurable space]] and a measure.
  
 
====Basic notions and constructions====
 
====Basic notions and constructions====
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Inner measure $\mu_*$ and outer measure $\mu^*$ are defined for all subsets $A\subset X$ by
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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\}\,;$
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$A$ is called a ''null'' (or ''negligible'') set if $\mu^*(A)=0$; in this case the complement $X\setminus A$ is called a set of ''full measure'', and one says that $x\notin A$ for ''almost all'' $x$ (in other words, ''almost everywhere'').

Revision as of 20:10, 18 February 2012

$\newcommand{\Om}{\Omega} \newcommand{\A}{\mathcal A} \newcommand{\B}{\mathcal B} \newcommand{\M}{\mathcal M} $ A measure space is a triple $(X,\A,\mu)$ where $X$ is a set, $\A$ a σ-algebra of its subsets, and $\mu:\A\to[0,+\infty]$ a measure. Thus, a measure space consists of a measurable space and a measure.

Basic notions and constructions

Inner measure $\mu_*$ and outer measure $\mu^*$ are defined for all subsets $A\subset X$ by

$ \mu_*(A) = \max\{\mu(B):B\in\A,B\subset A\}\,,\quad \mu^*(A) = \min\{\mu(B):B\in\A,B\supset A\}\,;$

$A$ is called a null (or negligible) set if $\mu^*(A)=0$; in this case the complement $X\setminus A$ is called a set of full measure, and one says that $x\notin A$ for almost all $x$ (in other words, almost everywhere).

How to Cite This Entry:
Boris Tsirelson/sandbox2. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Boris_Tsirelson/sandbox2&oldid=21186