Difference between revisions of "A pre-measure on a semi-ring may be extended uniquely to a pre-measure on a ring"
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(Created page with "{{Stub page|grade=A*|msg=Demote to grade A once fleshed out and grade C once (most of) a proof has been added}} <div style="display:none;"><m>\newcommand{\srmu}{\tilde{\mu}}\n...") |
(Added definitions of what the extended pre-measure should look like) |
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{{Stub page|grade=A*|msg=Demote to grade A once fleshed out and grade C once (most of) a proof has been added}} | {{Stub page|grade=A*|msg=Demote to grade A once fleshed out and grade C once (most of) a proof has been added}} | ||
| − | <div style="display:none;"><m>\newcommand{\srmu}{\tilde{\mu}}\newcommand{\rmu}{\bar{\mu}}</m></div> | + | <div style="display:none;"><m>\newcommand{\srmu}{\tilde{\mu}}\newcommand{\rmu}{\bar{\mu}}</m>{{Extra Maths}}</div> |
__TOC__ | __TOC__ | ||
==Statement== | ==Statement== | ||
| − | Given a [[pre-measure on a semi-ring]], {{M|\tilde{\mu}:\mathcal{F}\rightarrow\overline{\mathbb{R}_{\ge 0} } }} (that is a [[function]] whose domain is a [[semi-ring of sets]] that is [[countably additive]] with {{M|1=\tilde{\mu}(\emptyset)=0}}) then we may [[extend (function)|extend]] {{M|\srmu}} to a [[pre-measure]], {{M|\rmu:R(\mathcal{F})\rightarrow\overline{\mathbb{R}_{\ge 0} } }}<ref group="Note">Here {{M|R(X)}} denotes the ''[[ring of sets]]'' [[ring of sets generated by|generated by]] a collection of sets, {{M|X}}.</ref>; furthermore this extension is unique{{rMIAMRLS}}. | + | Given a [[pre-measure on a semi-ring]], {{M|\tilde{\mu}:\mathcal{F}\rightarrow\overline{\mathbb{R}_{\ge 0} } }} (that is a [[function]] whose domain is a [[semi-ring of sets]] that is [[countably additive]] with {{M|1=\tilde{\mu}(\emptyset)=0}}) then we may [[extend (function)|extend]] {{M|\srmu}} to a [[pre-measure]], {{M|\rmu:R(\mathcal{F})\rightarrow\overline{\mathbb{R}_{\ge 0} } }}<ref group="Note">Here {{M|R(X)}} denotes the ''[[ring of sets]]'' [[ring of sets generated by|generated by]] a collection of sets, {{M|X}}.</ref>; furthermore this extension is unique{{rMIAMRLS}}. The details follow: |
| + | * The ring generated by a semi-ring is exactly the set of all finite disjoint unions of elements from that semiring. | ||
| + | ** That is to say, {{M|1=R(\mathcal{F})=\left\{\left.\bigudot_{i=1}^nA_i\ \right\vert\ (A_i)_{i=1}^n\subseteq\mathcal{F}\right\} }} | ||
| + | ** so any {{M|A\in R(\mathcal{F}) }} can be written as {{M|1=A=\bigudot_{i=1}^n A_i}} for some ''[[finite]]'' [[sequence]] of [[pariwise disjoint]] sets, {{MSeq|A_i|i|1|n|in=\mathcal{F} }}<ref group="Note">I've mentioned it a few times but in case it isn't clear: | ||
| + | * '''For {{M|A\in R(\mathcal{F})}} we have {{M|1=A=\bigudot_{i=1}^nA_i}} for some finite sequence, {{MSeq|A_i|i|1|n|in=\mathcal{F} }}, note the elements of the sequence are in {{M|\mathcal{F} }}'''</ref> | ||
| + | * We define the induced pre-measure, {{M|\rmu:R(\mathcal{F})\rightarrow\overline{\mathbb{R}_{\ge 0} } }} as follows: | ||
| + | ** {{M|1=\rmu:\bigudot_{i=1}^nA_i\mapsto\sum_{i=1}^n\srmu(A_i)}}, and we claim this map is [[well-defined]] | ||
==Prerequisites== | ==Prerequisites== | ||
# [[the ring of sets generated by a semi-ring is the set containing the semi-ring and all finite disjoint unions]] | # [[the ring of sets generated by a semi-ring is the set containing the semi-ring and all finite disjoint unions]] | ||
Latest revision as of 22:16, 19 August 2016
Stub grade: A*
This page is a stub
This page is a stub, so it contains little or minimal information and is on a to-do list for being expanded.The message provided is:
Demote to grade A once fleshed out and grade C once (most of) a proof has been added
Statement
Given a pre-measure on a semi-ring, [ilmath]\tilde{\mu}:\mathcal{F}\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath] (that is a function whose domain is a semi-ring of sets that is countably additive with [ilmath]\tilde{\mu}(\emptyset)=0[/ilmath]) then we may extend [ilmath]\srmu[/ilmath] to a pre-measure, [ilmath]\rmu:R(\mathcal{F})\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath][Note 1]; furthermore this extension is unique[1]. The details follow:
- The ring generated by a semi-ring is exactly the set of all finite disjoint unions of elements from that semiring.
- That is to say, [ilmath]R(\mathcal{F})=\left\{\left.\bigudot_{i=1}^nA_i\ \right\vert\ (A_i)_{i=1}^n\subseteq\mathcal{F}\right\}[/ilmath]
- so any [ilmath]A\in R(\mathcal{F}) [/ilmath] can be written as [ilmath]A=\bigudot_{i=1}^n A_i[/ilmath] for some finite sequence of pariwise disjoint sets, [ilmath] ({ A_i })_{ i = 1 }^{ n }\subseteq \mathcal{F} [/ilmath][Note 2]
- We define the induced pre-measure, [ilmath]\rmu:R(\mathcal{F})\rightarrow\overline{\mathbb{R}_{\ge 0} } [/ilmath] as follows:
- [ilmath]\rmu:\bigudot_{i=1}^nA_i\mapsto\sum_{i=1}^n\srmu(A_i)[/ilmath], and we claim this map is well-defined
Prerequisites
Proof
Grade: A
This page requires one or more proofs to be filled in, it is on a to-do list for being expanded with them.
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The message provided is:
The message provided is:
I need to create the pre-measure on a semi-ring page and the ring of sets generated by a semi-ring is the set containing the semi-ring and all finite disjoint unions page before proceeding. See page 39 in[1]
See also
Notes
- ↑ Here [ilmath]R(X)[/ilmath] denotes the ring of sets generated by a collection of sets, [ilmath]X[/ilmath].
- ↑ I've mentioned it a few times but in case it isn't clear:
- For [ilmath]A\in R(\mathcal{F})[/ilmath] we have [ilmath]A=\bigudot_{i=1}^nA_i[/ilmath] for some finite sequence, [ilmath] ({ A_i })_{ i = 1 }^{ n }\subseteq \mathcal{F} [/ilmath], note the elements of the sequence are in [ilmath]\mathcal{F} [/ilmath]
References
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