Event Space from Single Subset of Sample Space

Theorem
Let $\mathcal E$ be an experiment whose sample space is $\Omega$.

Let $\varnothing \subsetneqq A \subsetneqq \omega$.

Then $\Sigma := \left\{{\varnothing, A, \Omega \setminus A, \Omega}\right\}$ is an event space of $\mathcal E$.

Proof
From its definition:
 * Event Space Axiom $(ES \ 1)$:
 * $\Sigma \ne \varnothing$

thus fulfilling axiom $(ES \ 1)$.


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

From Set Difference with Empty Set is Self:
 * $\Omega \setminus \varnothing = \Omega \in \Sigma$

From Set Difference with Self is Empty Set:
 * $\Omega \setminus \Omega = \varnothing \in \Sigma$

By definition:
 * $\Omega \setminus A \in \Sigma$

From [[Relative Complement of Relative Complement:
 * $\Omega \setminus \left({\Omega \setminus A}\right) = A \in \Sigma$

Thus axiom $(ES \ 2)$ is fulfilled.


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

From Union with Empty Set:
 * $\forall X \in \Sigma: X \cup \varnothing = X \in \Sigma$

From Union with Superset is Superset:
 * $\forall X \in \Sigma: X \cup \Sigma = \Sigma \in \Sigma$

From Union is Idempotent:
 * $\forall X \in \Sigma: X \cup X = X \in \Sigma$

From Union with Relative Complement:
 * $A \cup \left({\Omega \setminus A}\right) = \Sigma \in \Sigma$

It follows that axiom $(ES \ 3)$ is fulfilled.