Classical Probability is Probability Measure

Theorem
The Classical Probability model is a Probability Measure.

Proof
We check all the Kolmogorov Axioms in turn:

First Axiom
From Empty Set Subset of All and from the definition of the Event Space:


 * $\varnothing \subseteq \Sigma \subseteq \Omega $

From Cardinality of Empty Set and Proper Subset has Fewer Elements:


 * $0 \le \#\left(\Sigma\right) \le \#\left(\Omega\right) $

Dividing all terms by $\#\left(\Omega\right)$:


 * $0 \le \dfrac {\#\left(\Sigma\right)}{\#\left(\Omega\right)} \le 1$

The middle term is the asserted definition of $\Pr \left(\cdot\right)$.

Second Axiom
By hypothesis,


 * $\Pr \left({\Omega}\right) = \dfrac {\#\left(\Omega\right)}{\#\left(\Omega\right)} = 1$.

Third Axiom
Follows from Cardinality is an Additive Function and the Inclusion-Exclusion Principle.