Difference between revisions of "Outer-measure"
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==For every [[pre-measure]]== | ==For every [[pre-measure]]== | ||
<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. | <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. | ||
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Definition
An outer-measure, [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] - non-negative
- [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 non-negative with the property: [ilmath]\mu^*(\emptyset)=0[/ilmath] defined on a hereditary [ilmath]\sigma[/ilmath]-ring
For every pre-measure
[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
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