2024-12-22

913: σ-Algebra Induced on Codomain of Map from Measurable Space

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definition of σ-algebra induced on codomain of map from measurable space

Topics


About: measure

The table of contents of this article

Starting Context

Target Context


  • The reader will have a definition of σ-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:
(S1,A1): ∈{ the measurable spaces }
S2: ∈{ the sets }
f: :S1→S2
∗A2: ={s⊆S2:f−1(s)∈A1}, ∈{ the σ -algebras of S2}
//

Conditions:
//

2: Note


Let us see that A2 is indeed a σ-algebra of S2.

1) S2∈A2: f−1(S2)=S1∈A1.

2) ∀a∈A2(S2∖a∈A2): f−1(S2∖a)=f−1(S2)∖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, =S1∖f−1(a), but f−1(a)∈A1, so, S1∖f−1(a)∈A1.

3) ∀s:N→A2(∪j∈Ns(j)∈A2): f−1(∪j∈Ns(j))=∪j∈Nf−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))∈A2, so, ∪j∈Nf−1(s(j))∈A1.

References
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