# Semi-ring of half-closed-half-open intervals

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## Definition

Let [ilmath] a:= ({ a_i })_{ i = 1 }^{ n }\subseteq \mathbb{R} [/ilmath][Note 1][Note 2] and [ilmath] b:= ({ b_i })_{ i = 1 }^{ n }\subseteq \mathbb{R} [/ilmath] be two finite sequences of the same length (namely [ilmath]n\in\mathbb{N} [/ilmath]), we define [ilmath]\[a,b\)[/ilmath], a half-open-half-closed rectangle in [ilmath]\mathbb{R}^n[/ilmath] as follows:

• [ilmath]\[a,b\):=[a_1,b_1)\times\cdots\times[a_n,b_n)\subset\mathbb{R}^n[/ilmath] where [ilmath][\alpha,\beta):=\{x\in\mathbb{R}\ \vert\ \alpha\le x < \beta\}[/ilmath][Convention 1]

We denote the collection of all such half-open-half-closed rectangles by [ilmath]\mathscr{J}^n[/ilmath], [ilmath]\mathscr{J}(\mathbb{R}^n)[/ilmath] or, provided the context makes the dimensions obvious, simply just [ilmath]\mathscr{J} [/ilmath]. Formally:

• [ilmath]\mathscr{J}^n:=\{\[a,b\)\ \vert\ a,b\in\mathbb{R}^n\}[/ilmath]

Furthermore, we claim [ilmath]\mathscr{J}^n[/ilmath] is a [[semi-ring of sets][. For a proof of this claim see "Proof of claims" below.

## Purpose

Probably the most important use case for this semi-ring is as the domain of a certain kind of pre-measure, namely a pre-measure on a semi-ring that serves as a precursor to the Lebesgue measure, which for the reader's curiosity we include the definition for here:

• [ilmath]\lambda^n:\mathscr{J}^n\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath] with [ilmath]\lambda^n:\[a,b\)\mapsto\prod_{i=1}^n(b_i-a_i)[/ilmath][Note 3]

In the case of [ilmath]\mathscr{J}^1[/ilmath] the Lebesgue measure is just the length of an interval, that is: [ilmath]\lambda^1:[a,b)\mapsto (b-a)[/ilmath] and for [ilmath]\mathscr{J}^2[/ilmath] it is the area of a rectangle, for [ilmath]\mathscr{J}^3[/ilmath] volume of a cuboid, and so forth.

We can then use the theorem: a pre-measure on a semi-ring may be extended uniquely to a pre-measure on a ring to get a normal pre-measure. Doing this is far easier than trying to define a pre-measure on the ring of sets generated by [ilmath]\mathscr{J}^n[/ilmath].

Once we have a pre-measure we can follow the usual path of extending pre-measures to measures

## Proof of claims

Recall the definition of a semi-ring of sets

A collection of sets, [ilmath]\mathcal{F} [/ilmath][Note 4] is called a semi-ring of sets if:

1. [ilmath]\emptyset\in\mathcal{F}[/ilmath]
2. [ilmath]\forall S,T\in\mathcal{F}[S\cap T\in\mathcal{F}][/ilmath]
3. [ilmath]\forall S,T\in\mathcal{F}\exists(S_i)_{i=1}^m\subseteq\mathcal{F}[/ilmath][ilmath]\text{ pairwise disjoint}[/ilmath][ilmath][S-T=\bigudot_{i=1}^m S_i][/ilmath][Note 5] - this doesn't require [ilmath]S-T\in\mathcal{F} [/ilmath] note, it only requires that their be a finite collection of disjoint elements whose union is [ilmath]S-T[/ilmath].

In order to prove this we will first show that [ilmath]\mathscr{J}^1[/ilmath] (the collection of half-open-half-closed intervals of the form [ilmath][a,b)\subset\mathbb{R} [/ilmath]) is a semi-ring. Then we shall use induction on [ilmath]n[/ilmath] to show it for all [ilmath]\mathscr{J}^n[/ilmath]