Event Space from Single Subset of Sample Space

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

Let $\O \subsetneqq A \subsetneqq \Omega$.

Then $\Sigma := \set {\O, A, \Omega \setminus A, \Omega}$ is an event space of $\EE$.

Proof
From its definition:
 * Event Space Axiom $(\text {ES} 1)$:
 * $\Sigma \ne \O$

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


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

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

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

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

From Relative Complement of Relative Complement:
 * $\Omega \setminus \paren {\Omega \setminus A} = A \in \Sigma$

Thus axiom $(\text {ES} 2)$ is fulfilled.


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

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

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:
 * $A \cup \paren {\Omega \setminus A} = \Sigma \in \Sigma$

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