definition of independent indexed set of sub-\(\sigma\)-algebras of probability 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 sub-\(\sigma\)-algebras of probability 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:
\( (M, A, \mu)\): \(\in \{\text{ the probability spaces }\}\)
\( J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(*\{A_j \in \{\text{ the sub- } \sigma \text{ -algebras of } A\}\}_{j \in J}\): \(\in \{\text{ the indexed sets }\}\)
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Conditions:
\(\forall \{a_j \in A_j\}_{j \in J} (\{a_j\}_{j \in J} \in \{\text{ the independent indexed sets of events }\})\)
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2: Note
Only any \(a \in A\) such that \(\mu (a) \in \{0, 1\}\) can be shared in any \(2\) \(A_j\) s, because when \(a \in A_j, A_l\), \(\{a, a\}\) is independent, so, \(\mu (a \cap a) = \mu (a) \mu (a)\), but the left hand side is \(\mu (a)\), so, \(\mu (a) \in \{0, 1\}\).
\(\Omega\) and \(\emptyset\) are always shared in all the \(A_j\) s, because \(\Omega\) and \(\emptyset\) need to be contained in each \(A_j\) in order for \(A_j\) to be a \(\sigma\)-algebra.
