Doctrine:Measure theory terminology

Stub grade: A

This page is a stub

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:

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

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

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

$\mu^*$-measurable must be said with respect to an outer measure ($\mu^*$) 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 $X$ a splicing set then all ambiguity goes away and the name reflects what it does. In a sense:

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

If there is such a thing as $\mu_*$-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

• $\mathcal{S}^*$ for the set of all (outer) splicing sets with respect to the outer-measure $\mu^*$ say, of the context.
• $\mathcal{S}_*$ for the set of all inner splicing sets with respect to the inner-measure $\mu_*$ 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)