definition of independent indexed set of measurable maps from probability space into same measurable space
Topics
About: measure space
The table of contents of this article
Starting Context
- The reader knows a definition of probability space.
- The reader knows a definition of independent indexed set of events of probability space.
Target Context
- The reader will have a definition of independent indexed set of measurable maps from probability space into same 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:
\( (M, A, \mu)\): \(\in \{\text{ the probability spaces }\}\)
\( (M', A')\): \(\in \{\text{ the measurable spaces }\}\)
\( J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(*\{f_j: M \to M' \in \{\text{ the measurable maps }\}\}_{j \in J}\): \(\in \{\text{ the indexed sets }\}\)
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Conditions:
\(\forall \{a'_j \in A'\}_{j \in J} (\{{f_j}^{- 1} (a'_j)\}_{j \in J} \in \{\text{ the independent indexed sets of events }\})\)
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2: Note
\(\{f_j: M \to M' \in \{\text{ the measurable maps }\}\}_{j \in J}\) is independent if and only if \(\{\sigma (f_j)\}_{j \in J}\) is independent, by the proposition that any indexed set of measurable maps from any probability space into any same measurable space is independent if and only if the indexed set of the sub-\(\sigma\)-algebras induced by the maps is independent.
