Power Set of Sample Space is Event Space/Proof 1

Theorem

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

Let $\powerset \Omega$ be the power set of $\Omega$.

Then $\powerset \Omega$ is an event space of $\EE$.

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 $(\text {ES} 1)$.

$\Box$

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 $(\text {ES} 2)$.

$\Box$

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 $(\text {ES} 3)$.

$\Box$

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

Hence the result.

$\blacksquare$