definition of \(\sigma\)-algebra induced on codomain of map from measurable space
Topics
About: measure
The table of contents of this article
Starting Context
- The reader knows a definition of \(\sigma\)-algebra of set.
Target Context
- The reader will have a definition of \(\sigma\)-algebra induced on codomain of map from 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:
\( (S_1, A_1)\): \(\in \{\text{ the measurable spaces }\}\)
\( S_2\): \(\in \{\text{ the sets }\}\)
\( f\): \(: S_1 \to S_2\)
\(*A_2\): \(= \{s \subseteq S_2: f^{-1} (s) \in A_1\}\), \(\in \{\text{ the } \sigma \text{ -algebras of } S_2\}\)
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Conditions:
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2: Note
Let us see that \(A_2\) is indeed a \(\sigma\)-algebra of \(S_2\).
1) \(S_2 \in A_2\): \(f^{-1} (S_2) = S_1 \in A_1\).
2) \(\forall a \in A_2 (S_2 \setminus a \in A_2)\): \(f^{-1} (S_2 \setminus a) = f^{-1} (S_2) \setminus f^{-1} (a)\), by the proposition that for any map, the preimage of any subset minus any subset is the preimage of the 1st subset minus the preimage of the 2nd subset, \(= S_1 \setminus f^{-1} (a)\), but \(f^{-1} (a) \in A_1\), so, \(S_1 \setminus f^{-1} (a) \in A_1\).
3) \(\forall s: \mathbb{N} \to A_2 (\cup_{j \in \mathbb{N}} s (j) \in A_2)\): \(f^{-1} (\cup_{j \in \mathbb{N}} s (j)) = \cup_{j \in \mathbb{N}} f^{-1} (s (j))\), by the proposition that for any map, the map preimage of any union of sets is the union of the map preimages of the sets, but \(f^{-1} (s (j)) \in A_2\), so, \(\cup_{j \in \mathbb{N}} f^{-1} (s (j)) \in A_1\).
