Event Space contains Sample Space

Theorem

Let $\EE$ be an experiment with a probability space $\struct {\Omega, \Sigma, \Pr}$.

The event space $\Sigma$ of $\EE$ has the property that:

$\Omega \in \Sigma$

where $\Omega$ is the sample space of $\EE$.

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

\(\ds \leadsto \ \ \)

\(\ds \Omega \setminus A\)

\(\in\)

\(\ds \Sigma\)

Definition of Event Space: Axiom $(\text {ES} 2)$

\(\ds \leadsto \ \ \)

\(\ds A \cup \paren {\Omega \setminus A}\)

\(\in\)

\(\ds \Sigma\)

Definition of Event Space: Axiom $(\text {ES} 3)$

\(\ds \leadsto \ \ \)

\(\ds \Omega\)

\(\in\)

\(\ds \Sigma\)

Union with Relative Complement

$\blacksquare$

Also see

• Elementary Properties of Event Space

Sources

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