Doctrine:Measure theory terminology

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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

Outline

Measures

[ilmath]\xymatrix{ \text{Partial measure on a semi-ring} \ar@/^5ex/[drr] & \\ \vdots \ar[rr] & & \text{Pre-measure} \ar@{<-}@<1ex>@/^1ex/[ll]+<0ex,-5ex> \ar@{<-}@<-1ex>@/_1ex/[ll]+<0ex,5ex> \ar@/_3ex/[dr] \ar@{.>}@/_5ex/[rr]_{\text{via} }& & \text{Measure} \\ \text{(other partial measures)} \ar@/_5ex/[urr] & & & \text{Outer measure} \ar@/_3ex/[ur] \\ {\underbrace{\hphantom{\text{space} }\hphantom{\text{(other partial measures)} }\hphantom{\text{space} } }_{\text{Collectively called "partial measures"} } } & & & { \begin{array}{c} {\bf\text{on hereditary }\sigma\text{-ring} }\\ {\bf\text{(or power-set if }\sigma\text{-ring is a }\sigma\text{ algebra)} } \end{array} }\\ {\bf\text{on some sort of set structure} } & & {\bf\text{on a ring} } & & {\bf\text{on a }\sigma\text{-ring} } }[/ilmath]

Notes / provisional page

Terminology

  1. First we set up something achievable, the usual measure to consider here is the Lebesgue measure on half-open-half-closed rectangles[Note 1].
  2. Ring of sets
  3. Pre-measure
  4. Pre-measurable space
  5. Sigma-ring (of sets)
  6. Measurable space
  7. Measure
  8. Extending pre-measures to outer-measures
  9. 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

  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)

Notes

  1. ↑ 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]
    Then [ilmath]\lambda^n:\mathcal{J}^n\rightarrow\overline{\mathbb{R}_{\ge_0} } [/ilmath] is simply [ilmath]\lambda^n:[\![a,b)\!)\mapsto\prod_{i\eq 1}^n(b_i-a_i)[/ilmath]
    • TODO: Link to page explaining process
  2. ↑ Not every set is outer-measurable unless [ilmath]\mathcal{H} [/ilmath] is the powerset of the "universal set" in question

References

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