Difference between revisions of "Outer splicing set"

Revision as of 22:04, 20 August 2016

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]}

A set [ilmath]X[/ilmath] 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 (which is a [ilmath]\sigma[/ilmath]-ring and a critical part of extending pre-measures to measures) for more information.

See also

• The [ilmath]\sigma[/ilmath]-ring of all outer splicing sets

Notes

1. ↑ 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
2. ↑ 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

1. ↑ Measure Theory - Paul R. Halmos