Difference between revisions of "Doctrine:Measure theory terminology/Proposals"
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===Splicing sets=== | ===Splicing sets=== | ||
| − | I propose that rather than | + | I propose that rather than {{plural|mu*-measurable set|s}} we instead use [[outer splicing sets]] or just [[splicing sets]]. Currently: |
* For an [[outer-measure]], {{M|\mu^*:\mathcal{H}\rightarrow\overline{\mathbb{R}_{\ge 0} } }} we call a set, {{M|X\in\mathcal{H} }}, {{M|\mu^*}}-measurable if: | * For an [[outer-measure]], {{M|\mu^*:\mathcal{H}\rightarrow\overline{\mathbb{R}_{\ge 0} } }} we call a set, {{M|X\in\mathcal{H} }}, {{M|\mu^*}}-measurable if: | ||
** {{M|1=\forall Y\in\mathcal{H}[\mu^*(Y)=\mu^*(Y-X)+\mu^*(Y\cap X)]}} | ** {{M|1=\forall Y\in\mathcal{H}[\mu^*(Y)=\mu^*(Y-X)+\mu^*(Y\cap X)]}} | ||
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* {{M|X}} is a set that allows you to "splice" (the measures of) {{M|Y-X}} and {{M|Y\cap X}} together in a way which preserves the measure of {{M|Y}}. That is, the sum of the measures of the spliced parts is the measure of {{M|Y}}. | * {{M|X}} is a set that allows you to "splice" (the measures of) {{M|Y-X}} and {{M|Y\cap X}} together in a way which preserves the measure of {{M|Y}}. That is, the sum of the measures of the spliced parts is the measure of {{M|Y}}. | ||
If there is such a thing as {{M|\mu_*}}-measurable sets for the [[inner-measure]] they can simply be called "inner splicing sets" although I doubt that'll be needed. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21:14, 20 August 2016 (UTC) | If there is such a thing as {{M|\mu_*}}-measurable sets for the [[inner-measure]] they can simply be called "inner splicing sets" although I doubt that'll be needed. [[User:Alec|Alec]] ([[User talk: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) [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21:29, 20 August 2016 (UTC) | ||
====Standard symbols==== | ====Standard symbols==== | ||
* {{M|\mathcal{S}^*}} for the set of all (outer) splicing sets with respect to the [[outer-measure]] {{M|\mu^*}} say, of the context. | * {{M|\mathcal{S}^*}} for the set of all (outer) splicing sets with respect to the [[outer-measure]] {{M|\mu^*}} say, of the context. | ||
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====Points to address==== | ====Points to address==== | ||
# Is there such a thing as "inner splicing sets"? | # Is there such a thing as "inner splicing sets"? | ||
| + | #* There ''does not'' appear to be a corresponding notion for {{plural|inner-measure|s}} however there are similar things (see page 61 of Halmos' measure theory) in play | ||
# Does "splicing set" arise anywhere else? | # 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. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21:14, 20 August 2016 (UTC)<noinclude> | #* 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. [[User:Alec|Alec]] ([[User talk:Alec|talk]]) 21:14, 20 August 2016 (UTC)<noinclude> | ||
Latest revision as of 21:29, 20 August 2016
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.
Contents
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[Note 1]. 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
- 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
- Does "splicing set" arise anywhere else?
Notes
- ↑ Not every set is outer-measurable unless [ilmath]\mathcal{H} [/ilmath] is the powerset of the "universal set" in question
References
