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 is Subset of All Sets and from the definitions of the event space and sample space:


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

From Cardinality of Empty Set and Cardinality of Subset of Finite Set:


 * $0 \le \card \Sigma \le \card \Omega$

Dividing all terms by $\card \Omega$:


 * $0 \le \dfrac {\card \Sigma} {\card \Omega} \le 1$

The middle term is the asserted definition of $\map \Pr {\, \cdot \,}$.

Second Axiom
By hypothesis:


 * $\map \Pr \Omega = \dfrac {\card \Omega} {\card \Omega} = 1$

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