Event Space contains Empty Set

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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 property that:

$\O \in \Sigma$

That is, the empty set is in the event space.


Proof

\(\ds \Sigma\) \(\ne\) \(\ds \O\) Definition of Event Space: Axiom $(\text {ES} 1)$
\(\ds \leadsto \ \ \) \(\, \ds \exists A: \, \) \(\ds A\) \(\in\) \(\ds \Sigma\) Definition of Empty Set
\(\ds \leadsto \ \ \) \(\ds A \setminus A\) \(\in\) \(\ds \Sigma\) Definition of Event Space: Axiom $(\text {ES} 2)$
\(\ds \leadsto \ \ \) \(\ds \O\) \(\in\) \(\ds \Sigma\) Set Difference with Self is Empty Set

$\blacksquare$


Also see


Sources