Sample Space is Union of All Distinct Simple Events

Theorem

Let $\EE$ be an experiment.

Let $\Omega$ denote the sample space of $\EE$.

Then $\Omega$ is the union of the set of simple events in $\EE$.

Proof

By Set is Subset of Itself:

$\Omega \subseteq \Omega$

That is, $\Omega$ is itself an event in $\EE$.

The result as an application of Non-Trivial Event is Union of Simple Events.

$\blacksquare$

Sources

• 1965: A.M. Arthurs: Probability Theory ... (previous) ... (next): Chapter $2$: Probability and Discrete Sample Spaces: $2.2$ Sample spaces and events