910: σ-Algebra of Set Generated by Set of Subsets

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definition of $\sigma$-algebra of set generated by set of subsets

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Topics

About: measure

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of $\sigma$-algebra of set.

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Target Context

• The reader will have a definition of $\sigma$-algebra of set generated by set of subsets.

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Orientation

There is a list of definitions discussed so far in this site.

There is a list of propositions discussed so far in this site.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$S^{\prime }$: $\in \{\text{ the sets }\}$
$S$: $\subseteq Pow{S}^{\prime }$
$\ast \sigma (S)$: $\in \{\text{ the }\sigma \text{ -algebras of }{S}^{\prime }\}$
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Conditions:
$S\subseteq \sigma (S)$
$\wedge$
$\mathrm{\forall }{A}^{\prime }\in \{\text{ the }\sigma \text{ -algebras of }{S}^{\prime }\}\text{ such that }S\subseteq A^{\prime }(\sigma (S)\subseteq A^{\prime })$
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2: Note

In other words, $\sigma (S)$ is the smallest $\sigma$-algebra that contains $S$.

$\sigma (S)$ is uniquely determined, because it is the intersection of all the $\sigma$-algebras of $S^{\prime }$ that contain $S$, while at least, $Pow{S}^{\prime }$ is such a one: the intersection is a $\sigma$-algebra of $S^{\prime }$ that contains $S$, by the proposition that for any set, the intersection of any $\sigma$-algebras is a $\sigma$-algebra; for each $A^{\prime }\in \{\text{ the }\sigma \text{ -algebras of }{S}^{\prime }\}\text{ such that }S\subseteq A^{\prime }$, $\sigma (S)\subseteq A^{\prime }$, because $A^{\prime }$ is a constituent of the intersection.

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References

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