Difference between revisions of "Outer-measure"
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(Created page with " ==Definition== <math>\mu^*=\text{Inf}\left\{\sum^\infty_{n=1}\mu(E_n)|E_n\in R\ \forall n,\ E\subset\bigcup^\infty_{n=1}E_n\right\}</math> {{Definition|Measure Theory}}") |
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==Definition== | ==Definition== | ||
| − | <math>\mu^*=\text{Inf}\left\{\sum^\infty_{n=1}\mu(E_n) | + | 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}}]] | ||
| + | ==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== | ||
| + | <references/> | ||
| + | {{Measure theory navbox|plain}} | ||
{{Definition|Measure Theory}} | {{Definition|Measure Theory}} | ||
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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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