definition of measurable map from measurable space into topological space
Topics
About: measure
About: topological space
The table of contents of this article
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.
Target Context
- The reader will have a definition of measurable map from measurable space into topological 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:
\( (S, A)\): \(\in \{\text{ the measurable spaces }\}\)
\( T\): \(\in \{\text{ the topological spaces }\}\) with topology \(O\)
\(*f\): \(: S \to T\)
//
Conditions:
\(\forall o \in O (f^{-1} (o) \in A)\)
//
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.
