# Event Space contains Empty Set

## 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

$\blacksquare$