Power Set of Sample Space is Event Space/Proof 2

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


 * $(1): \quad \mathcal P \left({\Omega}\right) \ne \varnothing$, that is, an event space can not be empty.


 * $(2): \quad$ 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 $\mathcal P \left({\Omega}\right)$.


 * $(3): \quad$ If $A_1, A_2, \ldots \in \mathcal P \left({\Omega}\right)$, then $\displaystyle \bigcup_{i \mathop = 1}^\infty A_i \in \mathcal P \left({\Omega}\right)$, 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.