Difference between revisions of "Trace sigma-algebra"
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(Created page with "Let {{M|E\subseteq X}}, then {{M|1=\mathcal{A}_E:=\mathcal{A}\cap E:=\{A\cap E\vert A\in\mathcal{A}\} }} {{Todo|Measures Integrals and Martingales - page 16}} {{Definition|Me...") |
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− | Let {{M| | + | {{Stub page|grade=B|More results would be good. Relation to pullback too}} |
− | {{ | + | __TOC__ |
− | + | ==Definition== | |
+ | Let {{M|(X,\mathcal{A})}} be a [[sigma-algebra|{{sigma|algebra}}]] and let {{M|Y\subseteq X}} be any [[subset]] of {{M|X}}, then we may construct a {{sigma|algebra}} on {{M|Y}} called the ''trace {{sigma|algebra}}'', {{M|\mathcal{A}_Y}} given by{{rMIAMRLS}}: | ||
+ | * {{M|1=\mathcal{A}_Y:=\left\{Y\cap A\ \vert A\in\mathcal{A}\right\} }} | ||
+ | '''Claim: ''' {{M|(Y,\mathcal{A}_Y)}} is a {{sigma|algebra}} | ||
+ | ==Proof of claims== | ||
+ | {{Begin Inline Theorem}} | ||
+ | '''[[Trace sigma-algebra/Proof of claim that it actually is a sigma-algebra|Claim 1]]: ''' that {{M|(Y,\mathcal{A}_Y)}} is indeed a [[sigma-algebra|{{sigma|algebra}}]] | ||
+ | {{Begin Inline Proof}} | ||
+ | {{:Trace sigma-algebra/Proof of claim that it actually is a sigma-algebra}} | ||
+ | {{End Proof}}{{End Theorem}} | ||
+ | ==References== | ||
+ | <references/> | ||
+ | {{Measure theory navbox|plain}} | ||
{{Definition|Measure Theory}} | {{Definition|Measure Theory}} |
Latest revision as of 12:00, 23 August 2018
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More results would be good. Relation to pullback too
Contents
[hide]Definition
Let (X,A) be a σ-algebra and let Y⊆X be any subset of X, then we may construct a σ-algebra on Y called the trace σ-algebra, AY given by[1]:
- AY:={Y∩A |A∈A}
Claim: (Y,AY) is a σ-algebra
Proof of claims
References
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