Elementary Properties of Event Space
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Theorem
Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$.
The event space $\Sigma$ of $\EE$ has the following properties:
Event Space contains Empty Set
- $\O \in \Sigma$
Event Space contains Sample Space
- $\Omega \in \Sigma$
Intersection of Events is Event
- $A, B \in \Sigma \implies A \cap B \in \Sigma$
Set Difference of Events is Event
- $A, B \in \Sigma \implies A \setminus B \in \Sigma$
Symmetric Difference of Events is Event
- $A, B \in \Sigma \implies A \ast B \in \Sigma$
Countable Intersection of Events is Event
- $\quad A_1, A_2, \ldots \in \Sigma \implies \ds \bigcap_{i \mathop = 1}^\infty A_i \in \Sigma$
In the above:
- $A \setminus B$ denotes set difference
- $A \symdif B$ denotes symmetric difference.