920: Subspace σ-Algebra of Subset of Measurable Space

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definition of subspace $\sigma$-algebra of subset of measurable space

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Topics

About: measurable space

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The table of contents of this article

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

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Starting Context

• The reader knows a definition of measurable space.

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Target Context

• The reader will have a definition of subspace $\sigma$-algebra of subset of measurable space.

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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:
$(M^{\prime },A^{\prime })$: $\in \{\text{ the measurable spaces }\}$
$M$: $\subseteq M^{\prime }$
$\ast A$: $=\{W^{\prime }\cap M|W^{\prime }\in A^{\prime }\}$
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Conditions:
//

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2: Natural Language Description

For any measurable space, $(M^{\prime },A^{\prime })$, and any subset, $M\subseteq M^{\prime }$, $A:=\{W^{\prime }\cap M|W^{\prime }\in A^{\prime }\}$

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3: Note

$A$ is indeed a $\sigma$-algebra of $M$: 1) $M=M^{\prime }\cap M\in A$, because $M^{\prime }\in A^{\prime }$; 2) for each $W\in A$, $M\setminus W\in A$, because $M\setminus W=M\setminus (W^{\prime }\cap M)=(M^{\prime }\setminus W^{\prime })\cap M$, by the proposition that the intersection of any set minus any set and any set is the intersection of the 1st set and the 3rd set minus the intersection of the 2nd set and the 3rd set, and $M^{\prime }\setminus W^{\prime }\in A^{\prime }$; 3) for each infinite sequence, $W_1,W_2,...$, where $W_j\in A$, ${\cup }_j{W}_j\in A$, because ${\cup }_j{W}_j={\cup }_j(W_j^{\prime }\cap M)=({\cup }_j{W}_j^{\prime })\cap M$, by the proposition that for any set, the intersection of the union of any possibly uncountable number of subsets and any subset is the union of the intersections of each of the subsets and the latter subset, and ${\cup }_j{W}_j^{\prime }\in A^{\prime }$.

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References

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