description/proof of that continuous map between topological spaces with Borel \(\sigma\)-algebras is measurable
Topics
About: measurable space
The table of contents of this article
Starting Context
- The reader knows a definition of continuous, topological spaces map.
- The reader knows a definition of Borel \(\sigma\)-algebra of topological space.
- The reader knows a definition of measurable map between measurable spaces.
- The reader admits the proposition that for any map between any measurable spaces, if the codomain \(\sigma\)-algebra is generated by any set of subsets and the preimage of each element of the set of subsets is measurable, the map is measurable.
Target Context
- The reader will have a description and a proof of the proposition that any continuous map between any topological spaces with the Borel \(\sigma\)-algebras is measurable.
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:
\((T_1, \sigma (O_1))\): \(\in \{\text{ the measurable spaces }\}\), with \(T_1\) any topological space with any topology, \(O_1\)
\((T_2, \sigma (O_2))\): \(\in \{\text{ the measurable spaces }\}\), with \(T_2\) any topological space with any topology, \(O_2\)
\(f\): \(: T_1 \to T_2\), \(\in \{\text{ the continuous maps }\}\)
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Statements:
\(f \in \{\text{ the measurable maps }\}\)
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2: Proof
Whole Strategy: Step 1: apply the proposition that for any map between any measurable spaces, if the codomain \(\sigma\)-algebra is generated by any set of subsets and the preimage of each element of the set of subsets is measurable, the map is measurable.
Step 1:
For each \(U \in O_2\), \(f^{-1} (U) \in O_1 \subseteq \sigma (O_1)\).
\(f\) is measurable, by the proposition that for any map between any measurable spaces, if the codomain \(\sigma\)-algebra is generated by any set of subsets and the preimage of each element of the set of subsets is measurable, the map is measurable.
