Difference between revisions of "Doctrine:Measure theory terminology"
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{{Stub page|grade=A|msg=Templates for doctrine pages and entries for the different stages (proposed, fast-track, accepted...) need to be created. For now however I just want to note splicing sets}} | {{Stub page|grade=A|msg=Templates for doctrine pages and entries for the different stages (proposed, fast-track, accepted...) need to be created. For now however I just want to note splicing sets}} | ||
+ | ==Terminology== | ||
+ | # First we set up something achievable, the usual measure to consider here is the [[Lebesgue measure]] on half-open-half-closed rectangles<ref group="Note">Let {{M|\mathcal{J}^n }} be the set of all "half-open half closed rectangles", ie: | ||
+ | * {{M|[\![a,b)\!)\in\mathcal{J}^n}} if {{M|1=a:=(a_1,\ldots,a_n)\subseteq\mathbb{R}^n}} and {{M|1=b:=(b_1,\ldots,b_n)\subseteq \mathbb{R}^n}} (we could use {{M|\mathbb{Q} }} instead of {{M|\mathbb{R} }}) and | ||
+ | ** {{M|[\![a,b)):\eq\prod_{i\eq 1}^n[a_i,b_i)\eq[a_1,b_1)\times\cdots\times[a_n,b_n)}} | ||
+ | Then {{M|\lambda^n:\mathcal{J}^n\rightarrow\overline{\mathbb{R}_{\ge_0} } }} is simply {{M|\lambda^n:[\![a,b)\!)\mapsto\prod_{i\eq 1}^n(b_i-a_i)}} | ||
+ | * {{XXX|Link to page explaining process}}</ref>. | ||
+ | # [[Ring of sets]] | ||
+ | # [[Pre-measure]] | ||
+ | # [[Pre-measurable space]] | ||
+ | # [[Sigma-ring]] (of sets) | ||
+ | # [[Measurable space]] | ||
+ | # [[Measure]] | ||
+ | # [[Extending pre-measures to outer-measures]] | ||
+ | # [[Outer splicing set]] | ||
+ | {{Todo|List more of the process}} | ||
==[[/Proposals|Proposals]]== | ==[[/Proposals|Proposals]]== | ||
{{/Proposals}} | {{/Proposals}} |
Revision as of 23:54, 19 December 2016
Contents
Terminology
- First we set up something achievable, the usual measure to consider here is the Lebesgue measure on half-open-half-closed rectangles[Note 1].
- Ring of sets
- Pre-measure
- Pre-measurable space
- Sigma-ring (of sets)
- Measurable space
- Measure
- Extending pre-measures to outer-measures
- Outer splicing set
TODO: List more of the process
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 2]. 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
- ↑ Let [ilmath]\mathcal{J}^n [/ilmath] be the set of all "half-open half closed rectangles", ie:
- [ilmath][\![a,b)\!)\in\mathcal{J}^n[/ilmath] if [ilmath]a:=(a_1,\ldots,a_n)\subseteq\mathbb{R}^n[/ilmath] and [ilmath]b:=(b_1,\ldots,b_n)\subseteq \mathbb{R}^n[/ilmath] (we could use [ilmath]\mathbb{Q} [/ilmath] instead of [ilmath]\mathbb{R} [/ilmath]) and
- [ilmath][\![a,b)):\eq\prod_{i\eq 1}^n[a_i,b_i)\eq[a_1,b_1)\times\cdots\times[a_n,b_n)[/ilmath]
- TODO: Link to page explaining process
- [ilmath][\![a,b)\!)\in\mathcal{J}^n[/ilmath] if [ilmath]a:=(a_1,\ldots,a_n)\subseteq\mathbb{R}^n[/ilmath] and [ilmath]b:=(b_1,\ldots,b_n)\subseteq \mathbb{R}^n[/ilmath] (we could use [ilmath]\mathbb{Q} [/ilmath] instead of [ilmath]\mathbb{R} [/ilmath]) and
- ↑ Not every set is outer-measurable unless [ilmath]\mathcal{H} [/ilmath] is the powerset of the "universal set" in question