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:

$\O \subseteq \Sigma \subseteq \Omega$

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 $\map \Pr {\, \cdot \,}$.

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$\Box$

Second Axiom

By hypothesis:

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

$\Box$

Third Axiom

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

$\blacksquare$

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