Semi-ring of sets
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Definition
A collection of sets, [ilmath]\mathcal{F} [/ilmath][Note 1] is called a semi-ring of sets if[1]:
- [ilmath]\emptyset\in\mathcal{F}[/ilmath]
- [ilmath]\forall S,T\in\mathcal{F}[S\cap T\in\mathcal{F}][/ilmath]
- [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 2] - 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].
Purpose
The main motivation for semi-rings (in Measure Theory at least) is to let us provide a pre-measure on a semi-ring (a kind of pre-measure[Note 3]) and then use a theorem to prove this can be extended to a normal pre-measure (a similar structure define on a ring of sets instead). Then we can apply extending pre-measures to outer-measures to obtain an outer-measure, all without going through the tedious task of defining a pre-measure on a ring and doing only the basics by defining it on a semi-ring.
Examples
- Lebesgue pre-measure on a semi-ring - in one dimension the semi-ring, [ilmath]\mathscr{J}^1[/ilmath], here is the collection of all half-open-half-closed intervals on the real line, [ilmath][a,b)\subset\mathbb{R} [/ilmath] (with [ilmath][a,b):=\{x\in\mathbb{R}\ \vert a\le x < r\}[/ilmath]) for [ilmath]a,b\in\mathbb{R} [/ilmath] with the convention that if then [ilmath][a,b)=\emptyset[/ilmath].
- Clearly, [ilmath]\emptyset\in\mathscr{J}^1[/ilmath]
- Let [ilmath]a,b,c,d\in\mathbb{R} [/ilmath], Suppose [ilmath]a<b[/ilmath] and [ilmath]c<d[/ilmath] (as if either interval is the empty set the result is trivial). Suppose they partially intersect with [ilmath]a<c[/ilmath] and [ilmath]b<d[/ilmath], then clearly [ilmath][a,b)\cap[c,d)=[c,b)[/ilmath], this is the most difficult case.
- Using the same variables, the "hardest" case is that of [ilmath]a<c<d<b[/ilmath] so we have [ilmath][a,b)-[c,d)[/ilmath] with [ilmath][c,d)[/ilmath] being inside [ilmath][a,b)[/ilmath], then: [ilmath][a,b)-[c,d)=[a,c)\cup[d,b)[/ilmath]. The other cases are easier still.
- Showing even [ilmath]\mathscr{J}^1[/ilmath] is a ring of sets is very tedious. If the reader cannot see this, he should try it. Where as defining pre-measure on a semi-ring instead is something we've already done most of the work for!
See also
- Pre-measure on a semi-ring (an instance of a pre-measure)
- A pre-measure on a semi-ring may be extended uniquely to a pre-measure on a ring
- Types of set algebras
Notes
- ↑ An F is a bit like an R with an unfinished loop and the foot at the right. "Semi Ring".
- ↑ 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].
- 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
- [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]
- ↑ Many authors consider a pre-measure to be something we can extend to a measure somehow. We do not use this. Instead we define a pre-measure as being a function with certain properties on a ring of sets. This is useful because a pre-measure, under this definition, is almost a measure. A ring of sets is closed under all the elementary set operations.
We also adopt the convention of calling anything that can be extended to either a pre-measure (and thus an outer-measure and later a measure) a pre-measure on [ilmath]X[/ilmath] where [ilmath]X[/ilmath] is say a semi-ring or something.
All we need to do is show the pre-measure on [ilmath]X[/ilmath] extends uniquely to a pre-measure to allow the theorems (extending pre-measures to measures) to yield us a measure.
References
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