Power Set of Sample Space is an Event Space

Theorem

Let $\mathcal E$ be an experiment whose sample space is $\Omega$.

Let $\mathcal P \left({\Omega}\right)$ be the power set of $\Omega$.

Then $\mathcal P \left({\Omega}\right)$ is an event space of $\mathcal E$.

Proof

For $\mathcal P \left({\Omega}\right)$ to be an event space of $\mathcal E$, it needs to fulfil the following properties:

• $\mathcal P \left({\Omega}\right) \ne \varnothing$, that is, an event space can not be empty.

• If $A \in \mathcal P \left({\Omega}\right)$, then $\complement_\Omega \left({A}\right) \in \mathcal P \left({\Omega}\right)$, that is, the complement of $A$ relative to $\Omega$, is also in $\mathcal P \left({\Omega}\right)$.

• If $A_1, A_2, \ldots \in \mathcal P \left({\Omega}\right)$, then $\displaystyle \bigcup_{i=1}^\infty A_i \in \mathcal P \left({\Omega}\right)$, that is, the union of any countable collection of elements of $\mathcal P \left({\Omega}\right)$ is also in $\mathcal P \left({\Omega}\right)$.

These all follow directly from Power Set is Sigma-Algebra.

$\blacksquare$

Sources

• Geoffrey Grimmett and Dominic Welsh: Probability: An Introduction (1986): $\S 1.2$: Example $4$