Difference between revisions of "Trace sigma-algebra"
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==Definition== | ==Definition== | ||
Revision as of 21:30, 19 April 2016
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"Trace", but it's a subspace concept! Find more references
Definition
Let [ilmath](X,\mathcal{A})[/ilmath] be a [ilmath]\sigma[/ilmath]-algebra and let [ilmath]Y\subseteq X[/ilmath] be any subset of [ilmath]X[/ilmath], then we may construct a [ilmath]\sigma[/ilmath]-algebra on [ilmath]Y[/ilmath] called the trace [ilmath]\sigma[/ilmath]-algebra, [ilmath]\mathcal{A}_Y[/ilmath] given by[1]:
- [ilmath]\mathcal{A}_Y:=\left\{Y\cap A\ \vert A\in\mathcal{A}\right\}[/ilmath]
Claim: [ilmath](Y,\mathcal{A}_Y)[/ilmath] is a [ilmath]\sigma[/ilmath]-algebra
Proof of claims
Claim 1: that [ilmath](Y,\mathcal{A}_Y)[/ilmath] is indeed a [ilmath]\sigma[/ilmath]-algebra
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Easy - just show definition
References
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