Non-Trivial Event is Union of Simple Events

Theorem

Let $\EE$ be an experiment.

Let $e$ be an event in $\EE$ such that $e \ne \O$.

That is, such that $e$ is non-trivial.

Then $e$ can be expressed as the union of a set of simple events in $\EE$.

Proof

By definition of event, $e$ is a subset of the sample space $\Omega$ of $\EE$.

By hypothesis:

$e \ne \O$

and so:

$\exists s \in \Omega: s \in e$

Let $S$ be the set defined as:

$S = \set {\set s: s \in e}$

By Union is Smallest Superset: Set of Sets it follows that:

$\ds \bigcup S \subseteq e$

Let $x \in e$.

Then by Singleton of Element is Subset:

$\set x \subseteq e$

and by definition of $S$ it follows that:

$\set x \in S$

and so by definition of set union:

$x \in \ds \bigcup S$

Thus we have:

$e \subseteq \ds \bigcup S$

The result follows by definition of set equality.

$\blacksquare$

Sources

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