definition of \(\sigma\)-algebra induced on domain of map into measurable space
Topics
About: measurable space
The table of contents of this article
Starting Context
- The reader knows a definition of \(\sigma\)-algebra of set.
- The reader knows a definition of map preimage of subset of codomain.
Target Context
- The reader will have a definition of \(\sigma\)-algebra induced on domain of map into 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:
\( S_1\): \(\in \{\text{ the sets }\}\)
\( (M_2, A_2)\): \(\in \{\text{ the measurable spaces }\}\)
\( f\): \(: S_1 \to M_2\)
\(*\sigma (f)\): \(= \{f^{-1} (a_2) \vert a_2 \in A_2\}\), \(\in \{\text{ the } \sigma \text{ -algebras of } S_1\}\)
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Conditions:
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2: Note
\(\sigma (f)\) is the smallest \(\sigma\)-algebra that makes \(f\) measurable, by the proposition that for any map from any set into any measurable space, the smallest \(\sigma\)-algebra of the domain that makes the map measurable is the set of the preimages of the measurable subsets of the codomain, so, especially, \(\sigma (f)\) is a \(\sigma\)-algebra.
