Difference between revisions of "Semi-ring of half-closed-half-open intervals"

Latest revision as of 00:23, 20 August 2016

[ilmath]\newcommand{\[}{[\![}\newcommand{\]}{]\!]}\newcommand{\(}{(\!(}\newcommand{\)}{)\!)} [/ilmath]

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]^[1] 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]^[1], [ilmath]\mathscr{J}(\mathbb{R}^n)[/ilmath]^[1] or, provided the context makes the dimensions obvious, simply just [ilmath]\mathscr{J} [/ilmath]^[1]. 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

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]

Conventions

1. ↑ For intervals in general we define the following:
   1. If [ilmath]\alpha\ge\beta[/ilmath] then [ilmath][\alpha,\beta)=\emptyset[/ilmath]
   2. If [ilmath]\{X_\alpha\}_{\alpha\in I} [/ilmath] is an arbitrary collection of sets where one or more of the [ilmath]X_\alpha[/ilmath] are the empty set, [ilmath]\emptyset[/ilmath], then:
      • [ilmath]\prod_{\alpha\in I}X_\alpha=\emptyset[/ilmath] (here [ilmath]\prod[/ilmath] denotes the Cartesian product)
   As such if there is one or more [ilmath]a_i[/ilmath] and [ilmath]b_i[/ilmath] such that [ilmath]a_i\ge b_i[/ilmath] then [ilmath]\[a,b\)=\emptyset[/ilmath]

Notes

1. ↑ Or equivalently, [ilmath]a\in\mathbb{R}^n[/ilmath], either way we get an [ilmath]n[/ilmath]-tuple of real numbers
2. ↑ The symbol [ilmath]\subset[/ilmath] could be used instead of [ilmath]\subseteq[/ilmath] but it doesn't matter, as:
   • [ilmath]\big[A\subset B\big]\implies\big[A\subseteq B\big][/ilmath]
3. ↑ Here [ilmath]\prod[/ilmath] denotes multiplication repeated over a range, in this case multiplication of real numbers
4. ↑ An F is a bit like an R with an unfinished loop and the foot at the right. "Semi Ring".
5. ↑ Usually the finite sequence [ilmath] ({ S_i })_{ i = m }^{ \infty }\subseteq \mathcal{F} [/ilmath] being pairwise disjoint is implied by the [ilmath]\bigudot[/ilmath] however here I have been explicit. To be more explicit we could say:
   • [ilmath]\forall S,T\in\mathcal{F}\exists(S_i)_{i=1}^m\subseteq\mathcal{F}\left[\underbrace{\big(\forall i,j\in\{1,\ldots,m\}\subset\mathbb{N}[i\ne j\implies S_i\cap S_j=\emptyset]\big)}_{\text{the }S_i\text{ are pairwise disjoint} }\overbrace{\wedge}^\text{and}\left(S-T=\bigcup_{i=1}^m S_i\right)\right][/ilmath]
     • Caution:The statement: [ilmath]\forall S,T\in\mathcal{F}\exists(S_i)_{i=1}^m\subseteq\mathcal{F}\left[\big(\forall i,j\in\{1,\ldots,m\}\subset\mathbb{N}[i\ne j\implies S_i\cap S_j=\emptyset]\big)\implies\left(S-T=\bigcup_{i=1}^m S_i\right)\right][/ilmath] is entirely different
       • In this statement we are only declaring that a finite sequence exists, and if it is NOT pairwise disjoint, then we may or may not have [ilmath]S-T=\bigcup_{i=1}^mS_i[/ilmath]. We require that they be pairwise disjoint AND their union be the set difference of [ilmath]S[/ilmath] and [ilmath]T[/ilmath].

References

1. ↑ ^{1.0 1.1 1.2 1.3 1.4} Measures, Integrals and Martingales - René L. Schilling