Power Set of Sample Space is Event Space/Proof 1

Proof
Let $\powerset \Omega := \Sigma$.


 * Event Space Axiom $(\text {ES} 1)$:

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

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

thus fulfilling axiom $(\text {ES} 1)$.


 * Event Space Axiom $(\text {ES} 2)$:

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 $(\text {ES} 2)$.


 * Event Space Axiom $(\text {ES} 3)$:

Let $\sequence {A_i}$ be a countably infinite sequence of sets in $\Sigma$.

Then from Power Set is Closed under Countable Unions:


 * $\ds \bigcup_{i \mathop \in \N} A_i \in \Sigma$

thus fulfilling axiom $(\text {ES} 3)$.

All the event space axioms are seen to be fulfilled by $\powerset \Omega$.

Hence the result.