Difference between revisions of "Sigma-algebra generated by"

Latest revision as of 22:02, 17 March 2016

This page is a dire page and is in desperate need of an update.

See Just what is in a generated $\sigma$ -algebra for examples

Theorem statement

Given a set $S\subseteq\mathcal{P}(\Omega)$ (where $\mathcal{P}(\Omega)$ denotes the power set of $\Omega$ ) there exists^[1] a smallest $\sigma$ -algebra which we denote $\sigma(S)$ such that:

$S\subseteq\sigma(S)$ where $\sigma(S)=\bigcap_{\mathcal{A}\subseteq\mathcal{P}(\Omega)\text{ is a }\sigma\text{-algebra}\wedge S\subseteq\mathcal{A}}\mathcal{A}$

We say:

$\sigma(S)$ the $\sigma$ -algebra generated by $S$
$S$ the generator of $\sigma(S)$

[Expand]
Proof:

We will prove:

First that there actually is a $\sigma$ -algebra that contains $S$
Then there is a smallest $\sigma$ -algebra that contains $S$

Proof:

As P(Ω) is a σ-algebra we know a sigma algebra containing S 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 $\sigma(S)\subseteq\mathcal{P}(\Omega)$
Using the intersection of $\sigma$ -algebras is a $\sigma$ -algebra (for an arbitrary indexing set)
- If the indexing set is all the $\sigma$ -algebras that contain $S$ then we see immediately that $\sigma(S)$ is the smallest $\sigma$ -algebra containing $S$

The statement is proved

References

Jump up ↑ Probability Theory - A Comprehensive Course - Second Edition - Achim Klenke

[PTACC-1] Jump up ↑ Probability Theory - A Comprehensive Course - Second Edition - Achim Klenke

[1]

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+{{Definition|Measure Theory}}
 {{Theorem Of|Measure Theory}}

Difference between revisions of "Sigma-algebra generated by"

Latest revision as of 22:02, 17 March 2016

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