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
- 1986: Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction ... (previous) ... (next): $1$: Events and probabilities: $1.2$: Outcomes and events