Difference between revisions of "Sigma-algebra generated by"
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Revision as of 14:07, 16 June 2015
See Just what is in a generated [ilmath]\sigma[/ilmath]-algebra for examples
Contents
Theorem statement
Given a set [ilmath]S\subseteq\mathcal{P}(\Omega)[/ilmath] (where [ilmath]\mathcal{P}(\Omega)[/ilmath] denotes the power set of [ilmath]\Omega[/ilmath]) there exists[1] a smallest [ilmath]\sigma[/ilmath]-algebra which we denote [ilmath]\sigma(S)[/ilmath] such that:
- [ilmath]S\subseteq\sigma(S)[/ilmath] where [math]\sigma(S)=\bigcap_{\mathcal{A}\subseteq\mathcal{P}(\Omega)\text{ is a }\sigma\text{-algebra}\wedge S\subseteq\mathcal{A}}\mathcal{A}[/math]
We say:
- [ilmath]\sigma(S)[/ilmath] the [ilmath]\sigma[/ilmath]-algebra generated by [ilmath]S[/ilmath]
- [ilmath]S[/ilmath] the generator of [ilmath]\sigma(S)[/ilmath]
Proof:
We will prove:
- First that there actually is a [ilmath]\sigma[/ilmath]-algebra that contains [ilmath]S[/ilmath]
- Then there is a smallest [ilmath]\sigma[/ilmath]-algebra that contains [ilmath]S[/ilmath]
Proof:
- As [ilmath]\mathcal{P}(\Omega)[/ilmath] is a [ilmath]\sigma[/ilmath]-algebra we know a sigma algebra containing [ilmath]S[/ilmath] exists.
- So the intersection in the definition is non empty.
- By The intersection of sets is a subset of each set we now see that [ilmath]\sigma(S)\subseteq\mathcal{P}(\Omega)[/ilmath]
- Using the intersection of [ilmath]\sigma[/ilmath]-algebras is a [ilmath]\sigma[/ilmath]-algebra (for an arbitrary indexing set)
- If the indexing set is all the [ilmath]\sigma[/ilmath]-algebras that contain [ilmath]S[/ilmath] then we see immediately that [ilmath]\sigma(S)[/ilmath] is the smallest [ilmath]\sigma[/ilmath]-algebra containing [ilmath]S[/ilmath]
The statement is proved
See also
References
- ↑ Probability Theory - A Comprehensive Course - Second Edition - Achim Klenke
