Probability Measure on Finite Sample Space

Theorem
Let $\Omega = \set {\omega_1, \omega_2, \ldots, \omega_n}$ be a finite set.

Let $\Sigma$ be a $\sigma$-algebra on $\Omega$.

Let $p_1, p_2, \ldots, p_n$ be non-negative real numbers such that:
 * $p_1 + p_2 + \cdots + p_n = 1$

Let $Q: \Sigma \to \R$ be the mapping defined as:


 * $\forall A \in \Sigma: \map Q A = \ds \sum_{i: \omega_i \in A} p_i$

Then $\struct {\Omega, \Sigma, Q}$ constitutes a probability space.

That is, $Q$ is a probability measure on $\struct {\Omega, \Sigma}$.