Doctrine:Measure theory terminology/Proposals

This is a sub page for making proposals to the measure theory terminology doctrine. New requests only must be placed here. Queries and suggestions must not be put here unless there is a consensus (and thus proposal) on how to deal with it.

• Be sure to sign any proposals.

Proposals

Splicing sets

I propose that rather than mu*-measurable sets we instead use outer splicing sets or just splicing sets. Currently:

• For an outer-measure, [ilmath]\mu^*:\mathcal{H}\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath] we call a set, [ilmath]X\in\mathcal{H} [/ilmath], [ilmath]\mu^*[/ilmath]-measurable if:
  • [ilmath]\forall Y\in\mathcal{H}[\mu^*(Y)=\mu^*(Y-X)+\mu^*(Y\cap X)][/ilmath]

[ilmath]\mu^*[/ilmath]-measurable must be said with respect to an outer measure ([ilmath]\mu^*[/ilmath]) and is very close to "outer measurable set" which would just be an set the outer measure assigns a measure to. However if we call [ilmath]X[/ilmath] a splicing set then all ambiguity goes away and the name reflects what it does. In a sense:

• [ilmath]X[/ilmath] is a set that allows you to "splice" (the measures of) [ilmath]Y-X[/ilmath] and [ilmath]Y\cap X[/ilmath] together in a way which preserves the measure of [ilmath]Y[/ilmath]. That is, the sum of the measures of the spliced parts is the measure of [ilmath]Y[/ilmath].

If there is such a thing as [ilmath]\mu_*[/ilmath]-measurable sets for the inner-measure they can simply be called "inner splicing sets" although I doubt that'll be needed. Alec (talk) 21:14, 20 August 2016 (UTC)

Inner vs outer splicing sets=

I propose that when we speak of just a splicing set it be considered as an outer one (unless the context implies otherwise, for example if only inner-measures are in play) Alec (talk) 21:29, 20 August 2016 (UTC)

Standard symbols

• [ilmath]\mathcal{S}^*[/ilmath] for the set of all (outer) splicing sets with respect to the outer-measure [ilmath]\mu^*[/ilmath] say, of the context.
• [ilmath]\mathcal{S}_*[/ilmath] for the set of all inner splicing sets with respect to the inner-measure [ilmath]\mu_*[/ilmath] say, of the context. Caution:Should such a definition make sense.

Points to address

1. Is there such a thing as "inner splicing sets"?
   • There does not appear to be a corresponding notion for inner-measures however there are similar things (see page 61 of Halmos' measure theory) in play
2. Does "splicing set" arise anywhere else?
   • Yes, but in a niche area to do with a proper subset of regular languages and used with string splicing. So "splicing set" would not be ambiguous. Alec (talk) 21:14, 20 August 2016 (UTC)