Whole Sample Space represents Certain Event

Theorem

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

The sample space $\Sigma$ represents an event which is certain.

Proof

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

Hence an outcome of $\EE$ is necessarily an element of $\Omega$.

That is, the probability that $\omega \in \Omega$ is $1$.

The result follows by definition of certain event.

Sources

• 1998: David Nelson: The Penguin Dictionary of Mathematics (2nd ed.) ... (previous) ... (next): event
• 2008: David Nelson: The Penguin Dictionary of Mathematics (4th ed.) ... (previous) ... (next): event