description/proof of that for probability space and independent indexed set of events, indexed set of complements of events is independent
Topics
About: measure space
The table of contents of this article
Starting Context
- The reader knows a definition of independent indexed set of events of probability space.
- The reader admits the proposition for any set, the intersection of the compliments of any possibly uncountable number of subsets is the complement of the union of the subsets.
- The reader admits the proposition that for any probability space and any independent indexed set of events, for any finite indexed subset of the indexed set, \(1\) minus the probability of the union of the indexed subset is the product of \(1\) minus the probabilities of the elements of the indexed subset.
Target Context
- The reader will have a description and a proof of the proposition that for any probability space and any independent indexed set of events, the indexed set of the complements of the events is independent.
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:
\((M, A, \mu)\): \(\in \{\text{ the probability spaces }\}\)
\(J\): \(\in \{\text{ the possibly uncountable index sets }\}\)
\(S\): \(= \{a_j \in A\}_{j \in J}\), \(\in \{\text{ the independent indexed sets of events }\}\)
\(\widetilde{S}\): \(= \{M \setminus a_j \in A \vert j \in J\}\)
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Statements:
\(\widetilde{S} \in \{\text{ the independent indexed sets of events }\}\)
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2: Proof
Whole Strategy: Step 0: let \(J^` \subseteq J\) be any finite subset; Step 1: see that \(\mu (\cap_{j \in J^`} (M \setminus a_j)) = 1 - \mu (\cup_{j \in J^`} a_j)\); Step 2: see that \(\prod_{j \in J^`} \mu (M \setminus a_j) = \prod_{j \in J^`} (1 - \mu (a_j))\); Step 3: conclude the proposition.
Step 0:
Let \(J^` \subseteq J\) be any finite subset.
The proposition is about \(\mu (\cap_{j \in J^`} (M \setminus a_j)) = \prod_{j \in J^`} \mu (M \setminus a_j)\), which we are going to see.
Step 1:
\(\mu (\cap_{j \in J^`} (M \setminus a_j)) = \mu (M \setminus \cup_{j \in J^`} a_j)\), by the proposition for any set, the intersection of the compliments of any possibly uncountable number of subsets is the complement of the union of the subsets, \(= \mu (M) - \mu (\cup_{j \in J^`} a_j) = 1 - \mu (\cup_{j \in J^`} a_j)\).
Step 2:
\(\prod_{j \in J^`} \mu (M \setminus a_j) = \prod_{j \in J^`} (\mu (M) - \mu (a_j)) = \prod_{j \in J^`} (1 - \mu (a_j))\).
Step 3:
By the proposition that for any probability space and any independent indexed set of events, for any finite indexed subset of the indexed set, \(1\) minus the probability of the union of the indexed subset is the product of \(1\) minus the probabilities of the elements of the indexed subset, \(1 - \mu (\cup_{j \in J^`} a_j) = \prod_{j \in J^`} (1 - \mu (a_j))\), so, \(\mu (\cap_{j \in J^`} (M \setminus a_j)) = \prod_{j \in J^`} \mu (M \setminus a_j)\).
So, \(\widetilde{S}\) is independent.
