definition of product \(\sigma\)-algebra
Topics
About: measurable space
The table of contents of this article
- Starting Context
- Target Context
- Orientation
- Main Body
- 1: Structured Description 1
- 2: Structured Description 2
- 3: Note
Starting Context
- The reader knows a definition of measurable space.
- The reader knows a definition of product set.
- The reader knows \(\sigma\)-algebra of set generated by set of subsets.
Target Context
- The reader will have a definition of product \(\sigma\)-algebra.
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.
Main Body
1: Structured Description 1
Here is the rules of Structured Description.
Entities:
\( J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\( \{(M_j, A_j) \vert j \in J\}\): \((M_j, A_j) \in \{\text{ the measurable spaces }\}\)
\( M\): \(= \times_{j \in J} M_j\)
\( S\): \(= \{\times_{j \in J} a_j \subseteq M \vert a_j \in A_j \text{ where only finite of } a_j \text{ s are not } M_j s\}\)
\(*A\): \(\in \{\text{ the } \sigma \text{ -algebras of } M\}\)
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Conditions:
\(A = \text{ the } \sigma \text{ -algebra generated by } S\)
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\(\times_{j \in J} a_j\) is regarded to be a subset of \(\times_{j \in J} M_j\), because for each \(p \in \times_{j \in J} a_j\), \(p\) is a function from \(J\) into \(\cup_{j \in J} a_j\) such that \(p (j) \in a_j\), which is regarded to be a function from \(J\) into \(\cup_{j \in J} M_j\) such that \(p (j) \in M_j\).
2: Structured Description 2
Here is the rules of Structured Description.
Entities:
\( \{(M_1, A_1), ..., (M_n, A_n)\}\): \((M_j, A_j) \in \{\text{ the measurable spaces }\}\)
\( M\): \(= M_1 \times ... \times M_n\)
\( S\): \(= \{a_1 \times ... \times a_n \subseteq M \vert a_j \in A_j\}\)
\(*A\): \(\in \{\text{ the } \sigma \text{ -algebras of } M\}\)
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Conditions:
\(A = \text{ the } \sigma \text{ -algebra generated by } S\)
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3: Note
Sometimes, some notations like \(A = \times_{j \in J} A_j\) and \(A = A_1 \times ... \times A_n\) are used, but they are somewhat misleading, because for example, \(A_1 \times ... \times A_n\) is a notation of product set, but \(A\) is not the product set.
