View source for The set of all mu*-measurable sets is a ring

{{DISPLAYTITLE:The set of all {{M|\mu^*}}-measurable sets is a ring}}{{Stub page|grade=A*}}
==Statement==
{{M|\mathcal{S} }}, [[The set of all mu*-measurable sets|the set of all {{M|\mu^*}} measurable sets]], is a [[ring of sets]]{{rMTH}}.
: Recall that given an [[outer-measure]], {{M|\mu^*:H\rightarrow\bar{\mathbb{R} }_{\ge 0} }}, where {{M|H}} is a [[hereditary sigma-ring|hereditary {{sigma|ring}}]] that we call a set, {{M|A\in H}} ''[[Mu*-measurable set|{{M|\mu^*}}-measurable]]'' if<ref name="MTH"/>:
:* {{M|1=\forall B\in H[\mu^*(B)=\mu^*(B\cap A)+\mu^*(B-A)]}}. 
:* See the page [[Mu*-measurable set|{{M|\mu^*}}-measurable set]] for more information.
==Proof==
{{Requires writeup}}
I've thought about it, we know:
# {{M|1=\forall A\in H[\mu^*(A)=\mu^*(A\cap E)+\mu^*(A-E)]}} and
# {{M|1=\forall A\in H[\mu^*(A)=\mu^*(A\cap F)+\mu^*(A-F)]}}
And want to show:
* {{M|\forall A,B\in \mathcal{S}[A\cup B\in \mathcal{S}]}}
* {{M|\forall A,B\in \mathcal{S}[A-B\in \mathcal{S}]}}
Consider ANY {{M|3}} sets, {{M|A,\ E}} and {{M|F}}. Rather than dealing with "complicated" and ''non-unique'' expressions (eg {{M|1=(A\cap E)-F=(A-F)\cap E}} which are the same despite looking different, and there's no canonical or natural form for these sets), let us instead define:<div style="float:right;">
{| class="wikitable" style="width:14em;" border="1"
|-
| style="border-right:0em solid #000;" rowspan="8" |
| style="border-bottom:0em solid #000;border-left:0em solid #000;border-right:0em solid #000" colspan="7" |
| style="border-left:0em solid #000;" rowspan="8" |
|-
| style="border-left:0em solid #000;border-top:0em solid #000;" rowspan="3" | 
| style="border-left:0.2em solid #000;border-top:0.2em solid #000; border-right:0.2em solid #000;border-bottom:0px solid #000;" colspan="4" | <center>A</center>
| colspan="2" rowspan="2" | 
|-
| style="border-left:0.2em solid #000; border-right:0.2em solid #000;border-top:0em solid #000;border-bottom:0px solid #000;" colspan="4" | {{M|\alpha}}
|-
| style="border-left:0.2em solid #000;border-right:0em solid #000;border-top:0em solid #000;" colspan="2" | 
| style="border-left:0.2em solid #0000FF; border-top:0.2em solid #0000FF;border-right:0.2em solid #000; border-bottom:0em solid #000;text-align:right;" colspan="2" | {{M|\beta}}
| style="border-top:0.2em solid #0000FF; border-right:0.2em solid #0000FF;border-bottom:0em solid #000;border-left:0em solid #000;" colspan="2" rowspan="2" |
|-
| style="border-left:0.2em solid #00FF00;border-top:0.2em solid #00FF00; border-right:0.2em solid #000; border-bottom:0em;" |
| style="border-top:0.2em solid #00FF00;border-right:0em solid #000;" colspan="2" | {{M|\delta}}
| style="border-left:0.2em solid #0000FF;border-top:0.2em solid #00FF00; border-right:0.2em solid #00FF00;" | {{M|\gamma}}
| style="border-top:0em solid #000;border-right:0.2em solid #000;" | 
|-
| style="border-left:0.2em solid #00FF00;border-top:0em solid #000;border-bottom:0em solid #000;" | <div style="min-height:1em;min-width:1em;"></div>
| style="border-top:0.2em solid #000;border-bottom:0em solid #000;border-left:0em solid #000;" | <div style="min-height:1em;min-width:1em;"></div>
| style="border-top:0.2em solid #000;border-bottom:0em solid #000;border-right:0.2em solid #0000FF;" | <div style="min-height:1em;min-width:1em;"></div>
| style="border-left:0.2em solid #000; border-top:0.2em solid #000;border-right:0.2em solid #00FF00;border-bottom:0.2em solid #0000FF;" rowspan="2" |
| style="border-top:0.2em solid #000;border-bottom:0.2em solid #0000FF;border-right:0em solid #000;" rowspan="2" | 
| style="border-right:0.2em solid #0000FF; border-bottom:0.2em solid #0000FF;border-top:0em solid #000;border-left:0em solid #000;" colspan="2" rowspan="2" | E
|-
| style="border-top:0em solid #000;border-left:0.2em solid #00FF00;border-bottom:0.2em solid #00FF00;border-right:0em solid #000;" colspan="2" rowspan="2" | F
| <div style="min-height:1em;min-width:1em;"></div>
|-
| colspan="2" | <div style="min-height:1em;min-width:1em;"></div>
| <div style="min-height:1em;min-width:1em;"></div>
| <div style="min-height:1em;min-width:1em;"></div>
| <div style="min-height:1em;min-width:1em;"></div>
|-
| colspan="9" style="border-top:0em solid #000;" |
|-
! colspan="9" | Ven diagram showing the regions (some cells still have borders, only the coloured ones matter)
|}
</div>
* {{M|1=\alpha\ :=\ A-(E\cup F)}}
* {{M|1=\beta\ :=\ (A\cap E)-F}}
* {{M|1=\gamma\ :=\ A\cap E\cap F}}
* {{M|1=\delta\ :=\ (A\cap F)-E}}
Note that {{M|1=\forall \Omega\in\{\alpha,\ \beta,\ \gamma,\ \delta\} }}{{M|1=\Big[\big(\mu^*(\Omega)=\mu^*(\Omega\cap E)+\mu^*(\Omega-E)\big)}}{{M|\wedge}}{{M|1=\big(\mu^*(\Omega)=\mu^*(\Omega\cap F)+\mu^*(\Omega-F)\big)\Big]}}, as any such {{M|\Omega}} is a [[subset]] of {{M|A}}. And we have the above for all {{M|A\in H}}. As {{M|H}} is a [[hereditary system of sets]], we have it for all subsets of a given {{M|A}} too.

As a matter of notation, we write {{M|\Omega_1\Omega_2}} for {{M|\Omega_1\cup\Omega_2}}, so for example {{M|1=\beta\gamma\delta=A\cap(E\cup F)}}
===Proof #1===
* Let {{M|E,F\in S}} be given
** Let {{M|A\in H}} be given. We wish to show {{M|1=\mu^*(A)=\mu^*(A\cap(E\cup F))+\mu^*(A-(E\cup F))}} or {{M|1=\mu^*(A)=\mu^*(\beta\gamma\delta)+\mu^*(\alpha)}}
*** Notice {{M|1=\mu^*(\beta\gamma\delta)=\mu^*(\beta\gamma\delta\cap E)+\mu^*(\beta\gamma\delta-E)}} 
**** By tidying up the sets, we see this is: {{M|1=\mu^*(\beta\gamma\delta)=\mu^*(\beta\gamma)+\mu^*(\delta)}}
*** So {{M|1=\mu^*(\beta\gamma\delta)=\mu^*(\beta\gamma)+\mu^*(\delta)}}
*** Notice {{M|1=\mu^*(A)=\mu^*(A\cap E)+\mu^*(A-E)}}, or {{M|1=\mu^*(A)=\mu^*(\beta\gamma)+\mu^*(\alpha\delta)}} 
**** Re-arranging this, we see {{M|1=\mu^*(\beta\gamma)=\mu^*(A)-\mu^*(\alpha\delta)}}
*** Substituting this in: {{M|1=\mu^*(\beta\gamma\delta)=\mu^*(A)-\mu^*(\alpha\delta)+\mu^*(\delta)}}
**** Notice we can use {{M|F}} to split the {{M|\alpha\delta}} into {{M|1=\alpha\delta\cap F=\delta}} and {{M|1=\alpha\delta-F=\alpha}}
**** So {{M|1=\mu^*(\alpha\delta)=\mu^*(\delta)+\mu^*(\alpha)}}
*** Substituting this back in we see: {{M|1=\mu^*(\beta\gamma\delta)=\mu^*(A)-\mu^*(\alpha)-\mu^*(\delta)+\mu^*(\delta)}}
*** Simplifying: {{M|1=\mu^*(\beta\gamma\delta)=\mu^*(A)-\mu^*(\alpha)}}
*** Rearranging: {{M|1=\mu^*(A)=\mu^*(\beta\gamma\delta)+\mu^*(\alpha)}}
*** However notice:
***# {{M|1=\beta\gamma\delta=A\cap(E\cup F)}} and
***# {{M|1=\alpha=A-(E\cup F)}}
*** So we have:
**** {{M|1=\mu^*(A)=\mu^*(A\cap(E\cup F))+\mu^*(A-(E\cup F))}}
** Since {{M|A\in H}} was arbitrary we have: {{M|1=\forall A\in H[\mu^*(A)=\mu^*(A\cap(E\cup F))+\mu^*(A-(E\cup F))]}} 
*** So {{M|E\cup F\in S}}
* Since {{M|E,F\in S}} were arbitrary, we have {{M|\forall E,F\in S[E\cup F\in S]}}
** As a formula: {{M|1=\forall E,F\in S\big[\forall A\in H[\mu^*(A)=\mu^*(A\cap(E\cup F))+\mu^*(A-(E\cup F))]\big]}} {{Caution|{{M|\forall}}s commute, hence the brackets, I believe (I scratched a quick proof somewhere) that this is equivalent to the formula without the outer set of {{M|[\ ]}} however nothing is given ''so far'' - hence the brackets}}
This completes the proof.
===Proof #2===
Gist is the same, I did this on paper:
* {{M|\mu^*(\alpha\gamma\delta)}} (as {{M|1=\alpha\gamma\delta=A-(E-F)}}), split using {{M|F}} to get {{M|\mu^*(\alpha)+\mu^*(\gamma\delta)}}
* We need a {{M|\gamma\delta}} term, but {{M|1=\gamma\delta=A\cap F}} so by def of {{M|F}}:
** {{M|1=\mu^*(\gamma\delta)=\mu^*(A)-\mu^*(\alpha\beta)}}
* Now: {{M|1=\mu^*(\alpha\gamma\delta)=\mu^*(A)-\mu^*(\alpha\beta)+\mu^*(\alpha)}}
* {{M|\alpha\beta}} can be split by {{M|F}}, so {{M|1=\mu^*(\alpha\beta)=\mu^*(\alpha)+\mu^*(\beta)}}, thus:
* {{M|1=\mu^*(\alpha\gamma\delta)=\mu^*(A)-\mu^*(\alpha)-\mu^*(\beta)+\mu^*(\alpha)}}
The result follows
<div style="clear:both;"></div>
==See also==
* [[The set of all mu*-measurable sets is a sigma-ring|The set of all {{M|\mu^*}}-measurable sets is a {{sigma|ring}}]]
* [[The restriction of an outer-measure to the set of all mu*-measurable sets is a measure|The restriction of an outer-measure to the set of all {{M|\mu^*}}-measurable sets is a measure]]

==References==
<references/>
{{Measure theory navbox|plain}}
{{Theorem Of|Measure Theory}}