Outer splicing set

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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:

1. Unite this with the mu*-measurable set page, possibly by redirecting it here
2. 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 of [ilmath]\mu^*[/ilmath] (perhaps just splicing set if the context allows) or a [ilmath]\mu^*[/ilmath]-measurable set provided it satisfies the following:

• [ilmath]\forall Y\in\mathcal{H}[\mu^*(Y)=\mu^*(Y-X)+\mu^*(Y\cap X)][/ilmath]

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

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