Power Set of Sample Space is Event Space/Proof 1

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
Let $\mathcal P \left({\Omega}\right) := \Sigma$.


 * Event Space Axiom $(ES \ 1)$:

From Empty Set is Subset of All Sets we have that $\varnothing \subseteq \Omega$.

By the definition of power set:
 * $\varnothing \in \Sigma$

thus fulfilling axiom $(ES \ 1)$.


 * Event Space Axiom $(ES \ 2)$:

Let $A \in \Sigma$.

Then by the definition of power set:
 * $A \subseteq \Omega$

From Set with Relative Complement forms Partition:
 * $\Omega \setminus A \subseteq \Omega$

and so by the definition of power set:
 * $\Omega \setminus A \in \Sigma$

thus fulfilling axiom $(ES \ 2)$.


 * Event Space Axiom $(ES \ 3)$:

etc.