View source for Semi-ring of sets

==[[/Definition|Definition]]==
{{/Definition}}

==Purpose==
The main motivation for semi-rings (in [[Measure Theory (subject)|Measure Theory]] at least) is to let us provide a [[pre-measure on a semi-ring]] (a kind of [[pre-measure]]<ref group="Note">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. 
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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 {{C|pre-measure on {{M|X}}}} where {{M|X}} is say a semi-ring or something. 
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All we need to do is show the pre-measure on {{M|X}} extends uniquely to a pre-measure to allow the theorems ([[extending pre-measures to measures]]) to yield us a measure.</ref>) 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, {{M|\mathscr{J}^1}}, here is the collection of all half-open-half-closed {{plural|interval|s}} on the [[real line]], {{M|[a,b)\subset\mathbb{R} }} (with {{M|1=[a,b):=\{x\in\mathbb{R}\ \vert a\le x < r\} }}) for {{M|a,b\in\mathbb{R} }} with the convention that if {{M|a >= b}} then {{M|1=[a,b)=\emptyset}}.
*# Clearly, {{M|\emptyset\in\mathscr{J}^1}}
*# Let {{M|a,b,c,d\in\mathbb{R} }}, Suppose {{M|a<b}} and {{M|c<d}} (as if either interval is the empty set the result is trivial). Suppose they partially intersect with {{M|a<c}} and {{M|b<d}}, then clearly {{M|1=[a,b)\cap[c,d)=[c,b)}}, this is the most difficult case.
*# Using the same variables, the "hardest" case is that of {{M|a<c<d<b}} so we have {{M|[a,b)-[c,d)}} with {{M|[c,d)}} being inside {{M|[a,b)}}, then: {{M|1=[a,b)-[c,d)=[a,c)\cup[d,b)}}. The other cases are easier still.
** Showing even {{M|\mathscr{J}^1}} 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]]
** [[Ring of sets]]
*** [[sigma-ring|{{sigma|ring}}]]
** [[Algebra of sets]]
*** [[sigma-algebra|{{sigma|algebra}}]]
==Notes==
<references group="Note"/>
==References==
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