Outermeasure
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Definition
An outermeasure, [ilmath]\mu^*[/ilmath] is a set function from a hereditary [ilmath]\sigma[/ilmath]ring, [ilmath]\mathcal{H} [/ilmath], to the (positive) extended real values, [ilmath]\bar{\mathbb{R} }_{\ge0} [/ilmath], that is^{[1]}:
 [ilmath]\forall A\in\mathcal{H}[\mu^*(A)\ge 0][/ilmath]  nonnegative
 [ilmath]\forall A,B\in\mathcal{H}[A\subseteq B\implies \mu^*(A)\le\mu^*(B)][/ilmath]  monotonic
 [ilmath] \forall ({ A_n })_{ n = 1 }^{ \infty }\subseteq \mathcal{H} [\mu^*(\bigcup_{n=1}^\infty A_n)\le\sum^\infty_{n=1}\mu^*(A_n)] [/ilmath]  countably subadditive
In words, [ilmath]\mu^*[/ilmath] is:
 an extended real valued countably subadditive set function that is monotonic and nonnegative with the property: [ilmath]\mu^*(\emptyset)=0[/ilmath] defined on a hereditary [ilmath]\sigma[/ilmath]ring
For every premeasure
[math]\mu^*=\text{Inf}\left.\left\{\sum^\infty_{n=1}\bar{\mu(E_n)}\right\vert E_n\in R\ \forall n,\ E\subset\bigcup^\infty_{n=1}E_n\right\}[/math] is an outer measure.
References
