definition of product measurable space
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
Starting Context
- The reader knows a definition of product \(\sigma\)-algebra.
Target Context
- The reader will have a definition of product measurable space.
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:
\( J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\( \{(M_j, A_j) \vert j \in J\}\): \((M_j, A_j) \in \{\text{ the measurable spaces }\}\)
\( M\): \(= \times_{j \in J} M_j\)
\( A\): \(= \text{ the product } \sigma \text{ -algebra of } M\)
\(*(M, A)\): \(\in \{\text{ the measurable spaces }\}\)
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Conditions:
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2: Structured Description 2
Here is the rules of Structured Description.
Entities:
\( \{(M_1, A_1), ..., (M_n, A_n)\}\): \((M_j, A_j) \in \{\text{ the measurable spaces }\}\)
\( M\): \(= M_1 \times ... \times M_n\)
\( A\): \(= \text{ the product } \sigma \text{ -algebra of } M\)
\(*(M, A)\): \(\in \{\text{ the measurable spaces }\}\)
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Conditions:
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