View source for Outer-measure/Definition

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==Definition of an [[outer-measure]]==
</noinclude>An ''outer-measure'', {{M|\mu^*}} is a [[set function]] from a [[hereditary sigma-ring|hereditary {{sigma|ring}}]], {{M|\mathcal{H} }}, to the (positive) [[extended real value|extended real values]], {{M|\bar{\mathbb{R} }_{\ge0} }}, that is{{rMTH}}:
* {{M|\forall A\in\mathcal{H}[\mu^*(A)\ge 0]}} - non-negative
* {{M|\forall A,B\in\mathcal{H}[A\subseteq B\implies \mu^*(A)\le\mu^*(B)]}} - [[monotonic]]
* {{MSeq|A_n|in=\mathcal{H}|pre=\forall|post=[\mu^*(\bigcup_{n=1}^\infty A_n)\le\sum^\infty_{n=1}\mu^*(A_n)]}} - [[countably subadditive]]
In words, {{M|\mu^*}} is:
* an ''[[extended real valued]]'' [[countably subadditive set function]] that is [[monotonic]] and non-negative with the property: {{M|1=\mu^*(\emptyset)=0}} defined on a [[hereditary sigma-ring|hereditary {{sigma|ring}}]]<noinclude>
==References==
<references/>
{{Definition|Measure Theory}}
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