Discrete Uniform Distribution gives rise to Probability Measure

Theorem
Let $\EE$ be an experiment.

Let the probability space $\struct {\Omega, \Sigma, \Pr}$ be defined as:


 * $\Omega = \set {\omega_1, \omega_2, \ldots, \omega_n}$


 * $\Sigma = \powerset \Omega$


 * $\forall A \in \Sigma: \map \Pr A = \dfrac 1 n \card A$

where:


 * $\powerset \Omega$ denotes the power set of $\Omega$


 * $\card A$ denotes the cardinality of $A$.

Then $\Pr$ is a probability measure on $\struct {\Omega, \Sigma}$.

Proof
From Power Set of Sample Space is Event Space we have that $\Sigma$ is an event space.

We check the axioms defining a probability measure:

Axiom $\text I$ is seen to be satisfied by the observation that the cardinality of a set is never negative.

Hence $\map \Pr A \ge 0$.

Then we have:

Axiom $\text {II}$ is thus seen to be satisfied.

Let $A = \set {\omega_{r_1}, \omega_{r_2}, \ldots, \omega_{r_k} }$ where $k = \card A$.

Then by Union of Set of Singletons:


 * $A = \set {\omega_{r_1} } \cup \set {\omega_{r_2} } \cup \cdots \cup \set {\omega_{r_k} }$

Hence:

Hence Axiom $\text {III}$ is thus seen to be satisfied.

All axioms are seen to be satisfied.

Hence the result.