# Doctrine:Measure theory terminology

**Stub grade: A**

## Contents

## Outline

### Measures

# Notes / provisional page

## 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

- There
- 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