915: Measurable Map from Measurable Space into Topological Space

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definition of measurable map from measurable space into topological space

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Topics

About: measure
About: topological space

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The table of contents of this article

• Starting Context
• Target Context
• Orientation
• Main Body
• 1: Structured Description
• 2: Note

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Starting Context

• The reader knows a definition of measurable space.
• The reader knows a definition of topological space.
• The reader knows a definition of map.

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Target Context

• The reader will have a definition of measurable map from measurable space into topological space.

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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.

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Main Body

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1: Structured Description

Here is the rules of Structured Description.

Entities:
$(S,A)$: $\in \{\text{ the measurable spaces }\}$
$T$: $\in \{\text{ the topological spaces }\}$ with topology $O$
$\ast f$: $:S\to T$
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Conditions:
$\mathrm{\forall }o\in O(f^{-1}(o)\in A)$
//

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2: Note

In fact, this concept is equivalent with 'measurable map from measurable space into topological space turned measurable space with the Borel $\sigma$-algebra': make $T$ the measurable space, $(T,\sigma (O))$, where $\sigma (O)$ is the Borel $\sigma$-algebra, then, any map, $f:S\to T$, is measurable in this concept if and only if $f$ is measurable from $(S,A)$ into $(T,\sigma (O))$.

That is because if $f$ is measurable from $(S,A)$ into $(T,\sigma (O))$, $f$ is measurable in this concept because $O\subseteq \sigma (O)$, and if $f$ is measurable in this concept, $f$ is measurable from $(S,A)$ into $(T,\sigma (O))$ by the proposition that for any map between any measurable spaces, if the preimage of each element of any generator of the codomain $\sigma$-algebra is measurable, the map is measurable.

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References

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