Power Set of Sample Space is Event Space/Proof 2

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 $$\mathcal{C}_\Omega \left({A}\right) \in \mathcal{P} \left({\Omega}\right)$$, 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 $$\bigcup_{i=1}^\infty A_i \in \mathcal{P} \left({\Omega}\right)$$, that is, the union of any countable collection of elements of $$\mathcal{P} \left({\Omega}\right)$$ is also in $$\mathcal{P} \left({\Omega}\right)$$.

Event Space is Non-Empty
By Empty Set Element of Power Set, we have that $$\varnothing \in \mathcal{P} \left({\Omega}\right)$$ and so $$\mathcal{P} \left({\Omega}\right) \ne \varnothing$$.

Relative Complement in Event Space
If $$A \in \mathcal{P} \left({\Omega}\right)$$, then from Power Set is Algebra of Sets and the definition of Algebra of Sets, we have that:
 * $$A \in \mathcal{P} \left({\Omega}\right) \implies \mathcal{C}_\Omega \left({A}\right) \in \mathcal{P} \left({\Omega}\right)$$

where $$\mathcal{C}_\Omega \left({A}\right)$$ is the relative complement of $$A$$ in $$\Omega$$.

Countable Union of Events in Event Space
Now we need to show that $$A_1, A_2, \ldots \in \mathcal{P} \left({\Omega}\right) \implies \bigcup_{i=1}^\infty A_i \in \mathcal{P} \left({\Omega}\right)$$, that is, the union of any countable collection of elements of $$\mathcal{P} \left({\Omega}\right)$$ is also in $$\mathcal{P} \left({\Omega}\right)$$.