View source for Outer splicing set

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# Unite this with the [[mu*-measurable set]] page, possibly by redirecting it here
# Make it clear that is is [[:Category:Invented terminology|invented terminology]] and what the usual terms are
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: This page is a result of the [[Doctrine:Measure theory terminology]] document. It is not a well known term. [[mu*-measurable set|{{M|\mu*}}-measurable set]] redirects here.
:* '''Be sure to understand that this is usually called a "μ<sup>*</sup>-measurable set"'''
==Definition==
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>
==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==
* [[The sigma-ring of all outer splicing sets|The {{sigma|ring}} of all outer splicing sets]]
==Notes==
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==References==
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