Difference between revisions of "Outer splicing set"
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Let {{M|\mu^*:\mathcal{H}\rightarrow\overline{\mathbb{R}_{\ge 0} } }} be an [[outer-measure]]. A set, {{M|X\in\mathcal{H} }}, is said to be an ''outer splicing set''<ref group="Note">This is an [[:Category:Invented terminology|invented term]]. This is only ever done with good reason and will never cause ambiguity. The reader should see to it he is aware of the existing and more common terms</ref> of {{M|\mu^*}} (perhaps just ''splicing set'' if the context allows) or a ''{{M|\mu^*}}-measurable set''{{rMTH}} provided it satisfies the following: | Let {{M|\mu^*:\mathcal{H}\rightarrow\overline{\mathbb{R}_{\ge 0} } }} be an [[outer-measure]]. A set, {{M|X\in\mathcal{H} }}, is said to be an ''outer splicing set''<ref group="Note">This is an [[:Category:Invented terminology|invented term]]. This is only ever done with good reason and will never cause ambiguity. The reader should see to it he is aware of the existing and more common terms</ref> of {{M|\mu^*}} (perhaps just ''splicing set'' if the context allows) or a ''{{M|\mu^*}}-measurable set''{{rMTH}} provided it satisfies the following: | ||
* {{M|1=\forall Y\in\mathcal{H}[\mu^*(Y)=\mu^*(Y-X)+\mu^*(Y\cap X)]}}<ref group="Note">Some authors, for example Halmos, abuse notation quite a lot. For example Halmos gives a great abuse of notation here, by writing {{M|B\cap A'}} (where {{M|A'}} denotes the [[complement]] of {{M|A}}), of course in a [[ring of sets]] (sigma or not) we do not have a complementation operation, only set subtraction</ref> | * {{M|1=\forall Y\in\mathcal{H}[\mu^*(Y)=\mu^*(Y-X)+\mu^*(Y\cap X)]}}<ref group="Note">Some authors, for example Halmos, abuse notation quite a lot. For example Halmos gives a great abuse of notation here, by writing {{M|B\cap A'}} (where {{M|A'}} denotes the [[complement]] of {{M|A}}), of course in a [[ring of sets]] (sigma or not) we do not have a complementation operation, only set subtraction</ref> | ||
| − | + | ==Terminology and purpose== | |
| − | + | We call such a set a ''splicing set'' because it has the property that that "splicing" together {{M|Y-X}} and {{M|Y\cap X}} is exactly additive on the [[outer-measure]] {{M|\mu^*}}. Be aware that traditionally such sets are called [[mu*-measurable set|{{M|\mu^*}}-measurable sets]], as mentioned above. | |
| + | |||
| + | The collection of all ''outer splicing sets'' of {{M|\mu^*}} is usually denoted {{M|\mathcal{S}^*}} (see [[the set of all outer splicing sets]]) and furthermore this collection is a [[sigma-ring|{{sigma|ring}}]]. This observation is a critical part of [[extending pre-measures to measures]]. | ||
| + | |||
==See also== | ==See also== | ||
* [[The sigma-ring of all outer splicing sets|The {{sigma|ring}} of all outer splicing sets]] | * [[The sigma-ring of all outer splicing sets|The {{sigma|ring}} of all outer splicing sets]] | ||
Latest revision as of 22:09, 20 August 2016
Stub grade: A
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Still to do:
- Unite this with the mu*-measurable set page, possibly by redirecting it here
- Make it clear that is is invented terminology and what the usual terms are
- This page is a result of the Doctrine:Measure theory terminology document. It is not a well known term. [ilmath]\mu*[/ilmath]-measurable set redirects here.
- Be sure to understand that this is usually called a "μ*-measurable set"
Definition
Let [ilmath]\mu^*:\mathcal{H}\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath] be an outer-measure. A set, [ilmath]X\in\mathcal{H} [/ilmath], is said to be an outer splicing set[Note 1] of [ilmath]\mu^*[/ilmath] (perhaps just splicing set if the context allows) or a [ilmath]\mu^*[/ilmath]-measurable set[1] provided it satisfies the following:
- [ilmath]\forall Y\in\mathcal{H}[\mu^*(Y)=\mu^*(Y-X)+\mu^*(Y\cap X)][/ilmath][Note 2]
Terminology and purpose
We call such a set a splicing set because it has the property that that "splicing" together [ilmath]Y-X[/ilmath] and [ilmath]Y\cap X[/ilmath] is exactly additive on the outer-measure [ilmath]\mu^*[/ilmath]. Be aware that traditionally such sets are called [ilmath]\mu^*[/ilmath]-measurable sets, as mentioned above.
The collection of all outer splicing sets of [ilmath]\mu^*[/ilmath] is usually denoted [ilmath]\mathcal{S}^*[/ilmath] (see the set of all outer splicing sets) and furthermore this collection is a [ilmath]\sigma[/ilmath]-ring. This observation is a critical part of extending pre-measures to measures.
See also
Notes
- ↑ This is an invented term. This is only ever done with good reason and will never cause ambiguity. The reader should see to it he is aware of the existing and more common terms
- ↑ Some authors, for example Halmos, abuse notation quite a lot. For example Halmos gives a great abuse of notation here, by writing [ilmath]B\cap A'[/ilmath] (where [ilmath]A'[/ilmath] denotes the complement of [ilmath]A[/ilmath]), of course in a ring of sets (sigma or not) we do not have a complementation operation, only set subtraction
References
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