Difference between revisions of "Outer-measure"

Revision as of 23:13, 15 March 2015 (view source) Alec (Talk | contribs) (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}}")		Revision as of 17:43, 8 April 2016 (view source) Alec (Talk | contribs) m Newer edit →
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−		+	{{Stub page|Saving work, not even a stub yet!}}
	==Definition==		==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>	+	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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Revision as of 17:43, 8 April 2016

Definition

An outer-measure, $\mu^*$ is a set function from a hereditary $\sigma$-ring, $\mathcal{H}$, to the (positive) extended real values, $\bar{\mathbb{R} }_{\ge0}$, that is^[1]:

• $\forall A\in\mathcal{H}[\mu^*(A)\ge 0]$ - non-negative
• $\forall A,B\in\mathcal{H}[A\subseteq B\implies \mu^*(A)\le\mu^*(B)]$ - monotonic
• $\forall ({ A_n })_{ n = 1 }^{ \infty }\subseteq \mathcal{H} [\mu^*(\bigcup_{n=1}^\infty A_n)\le\sum^\infty_{n=1}\mu^*(A_n)]$ - countably subadditive

In words, $\mu^*$ is:

• an extended real valued countably subadditive set function that is monotonic and non-negative with the property: $\mu^*(\emptyset)=0$ defined on a hereditary $\sigma$-ring

For every pre-measure

$$\mu^*=\text{Inf}\left.\left\{\sum^\infty_{n=1}\bar{\mu(E_n)}\right\vert E_n\in R\ \forall n,\ E\subset\bigcup^\infty_{n=1}E_n\right\}$$ is an outer measure.

References

1. Jump up ↑ Measure Theory - Paul R. Halmos

v • d • e   Measure Theory Overview of the basic and important objects studied in measure theory Set-structures ring of sets, algebra of sets, $\sigma$-ring, $\sigma$-algebra, hereditary $\sigma$-ring measurable space Measures Pre-measure $\leadsto$ outer-measure $\leadsto$ measure (see Extending pre-measures to measures (via: Extending pre-measures to outer-measures $\rightarrow$ "Restricting outer-measures to measures" maybe?). Alternatively: inner-measure can be used. measure space Functions measurable map, simple function (standard representation) Spaces Integrals Integrating a simple function $\leadsto$ Integral of a positive function $\leadsto$ Integrating all measurable functions