Difference between revisions of "Trace sigma-algebra"
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==Definition== | ==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}}: | 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}}: | ||
Revision as of 11:59, 23 August 2018
Stub grade: B
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More results would be good. Relation to pullback too
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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