View source for Semi-ring of half-closed-half-open intervals

{{M|\newcommand{\[}{[\![}\newcommand{\]}{]\!]}\newcommand{\(}{(\!(}\newcommand{\)}{)\!)} }}
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==Definition==
Let {{MSeq|a_i|i|1|n|in=\mathbb{R}|pre=a:=}}<ref group="Note">Or equivalently, {{M|a\in\mathbb{R}^n}}, either way we get an {{n|[[tuple]]}} of real numbers</ref><ref group="Note">The symbol {{M|\subset}} could be used instead of {{M|\subseteq}} but it doesn't matter, as:
* {{M|\big[A\subset B\big]\implies\big[A\subseteq B\big]}}</ref> and {{MSeq|b_i|i|1|n|in=\mathbb{R}|pre=b:=}} be two ''[[finite]]'' {{plural|sequence|s}} of the same length (namely {{M|n\in\mathbb{N} }}), we define {{M|1=\[a,b\)}}, a ''half-open-half-closed rectangle in {{M|\mathbb{R}^n}}''{{rMIAMRLS}} as follows:
* {{M|1=\[a,b\):=[a_1,b_1)\times\cdots\times[a_n,b_n)\subset\mathbb{R}^n}} where {{M|1=[\alpha,\beta):=\{x\in\mathbb{R}\ \vert\ \alpha\le x < \beta\} }}<ref group="Convention">For [[interval|intervals]] in general we define the following:
# If {{M|\alpha\ge\beta}} then {{M|1=[\alpha,\beta)=\emptyset}}
# If {{M|\{X_\alpha\}_{\alpha\in I} }} is an [[arbitrary collection]] of [[set|sets]] where one or more of the {{M|X_\alpha}} are the [[empty set]], {{M|\emptyset}}, then:
#* {{M|1=\prod_{\alpha\in I}X_\alpha=\emptyset}} (here {{M|\prod}} denotes the [[Cartesian product]])
As such if there is one or more {{M|a_i}} and {{M|b_i}} such that {{M|a_i\ge b_i}} then {{M|1=\[a,b\)=\emptyset}}</ref>
We denote the collection of all such ''half-open-half-closed rectangles'' by {{M|\mathscr{J}^n}}<ref name="MIAMRLS"/>, {{M|\mathscr{J}(\mathbb{R}^n)}}<ref name="MIAMRLS"/> or, provided the context makes the dimensions obvious, simply just {{M|\mathscr{J} }}<ref name="MIAMRLS"/>. Formally:
* {{M|1=\mathscr{J}^n:=\{\[a,b\)\ \vert\ a,b\in\mathbb{R}^n\} }} 
Furthermore, we claim {{M|\mathscr{J}^n}} is a [[semi-ring of sets][. For a proof of this claim see [[Semi-ring_of_half-closed-half-open_intervals#Proof of claims|"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:
* {{M|\lambda^n:\mathscr{J}^n\rightarrow\overline{\mathbb{R}_{\ge 0} } }} with {{M|1=\lambda^n:\[a,b\)\mapsto\prod_{i=1}^n(b_i-a_i)}}<ref group="Note">Here {{M|\prod}} denotes multiplication repeated over a range, in this case multiplication of real numbers</ref>
In the case of {{M|\mathscr{J}^1}} the Lebesgue measure is just the length of an interval, that is: {{M|1=\lambda^1:[a,b)\mapsto (b-a)}} and for {{M|\mathscr{J}^2}} it is the area of a rectangle, for {{M|\mathscr{J}^3}} 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]] {{M|\mathscr{J}^n}}.


Once we have a [[pre-measure]] we can follow the usual path of [[extending pre-measures to measures]]
==Proof of claims==
{{Begin Notebox}}
Recall the definition of a [[semi-ring of sets]]
{{Begin Notebox Content}}
{{:Semi-ring of sets/Definition}}
{{End Notebox Content}}{{End Notebox}}
In order to prove this we will first show that {{M|\mathscr{J}^1}} (the collection of half-open-half-closed intervals of the form {{M|[a,b)\subset\mathbb{R} }}) is a semi-ring. Then we shall use induction on {{M|n}} to show it for all {{M|\mathscr{J}^n}}
{{Requires proof|grade=B|msg=It's a straightforward and fairly easy proof, but it isn't trivial. As such it is not marked as an easy proof, but nor does it have a high grade. See page 18 in [[Books:Measures, Integrals and Martingales - René L. Schilling|Measures, Integrals and Martingales - René L. Schilling]] if stuck.}}
==Conventions==
<references group="Convention"/>
==Notes==
<references group="Note"/>
==References==
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