Classical Probability is Probability Measure

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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:

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

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 $\Pr \left({\cdot}\right)$.



$\Box$


Second Axiom

By hypothesis:

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

$\Box$


Third Axiom

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

$\blacksquare$