Elementary Properties of Event Space

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Theorem

Let $\mathcal E$ be an experiment with a probability space $\left({\Omega, \Sigma, \Pr}\right)$.


The event space $\Sigma$ of $\mathcal E$ 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
$A \ast B$ denotes symmetric difference.