definition of Euclidean measurable space
Topics
About: measurable space
The table of contents of this article
Starting Context
- The reader knows a definition of Euclidean topological space.
- The reader knows a definition of Borel \(\sigma\)-algebra of topological space.
Target Context
- The reader will have a definition of Euclidean 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
Here is the rules of Structured Description.
Entities:
\( \mathbb{R}^d\): \(= \text{ the Euclidean topological space }\)
\( B (\mathbb{R}^d)\): \(= \text{ the Borel } \sigma \text{ -algebra of } \mathbb{R}^d\)
\(*(\mathbb{R}^d, B (\mathbb{R}^d))\): \(\in \{\text{ the measurable spaces }\}\)
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Conditions:
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2: Note
\((\mathbb{R}^d, B (\mathbb{R}^d))\) is \(\mathbb{R}^d\) with the product \(\sigma\)-algebra of \(B (\mathbb{R})\), by the proposition that the Borel \(\sigma\)-algebra of the \(d\)-dimensional Euclidean topological space is the product \(\sigma\)-algebra of the Borel \(\sigma\)-algebras of the \(1\)-dimensional Euclidean topological space.
