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:
- $0 \le \# \left({\Sigma}\right) \le \# \left({\Omega}\right)$
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 $\Pr \left({\cdot}\right)$.
Link to a page expanding on that statement -- perhaps better to say that it's the definition of the image of Pr?
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$\Box$
Second Axiom
- $\Pr \left({\Omega}\right) = \dfrac {\# \left({\Omega}\right)} {\# \left({\Omega}\right)} = 1$
$\Box$
Third Axiom
Follows from Cardinality is Additive Function and the corollary to the Inclusion-Exclusion Principle.
$\blacksquare$
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