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 definitions of the event space and sample 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 Corollary to the Inclusion-Exclusion Principle.