\begin{definition}[Definition:Discrete Probability Measure]
Let $\Omega$ be a countable set.
Let $\powerset \Omega$ be its power set, regarded as a $\sigma$-algebra.
Let $\paren {p_\omega}_{\omega \mathop \in \Omega} \subseteq \closedint 0 1$ be a subset of the closed unit interval in $\R$, indexed by $\Omega$.
Suppose that $\ds \sum_{\omega \mathop \in \Omega} p_\omega = 1$.
{{explain|link to definition of sum over countable set}}
The '''discrete probability measure on $\Omega$''', denoted $P$, is the mapping defined by:
:$\ds P: \powerset \Omega \to \overline \R, \map P S = \sum_{\omega \mathop \in \Omega} p_\omega \map {\delta_\omega} S$
where $\overline \R$ denotes the extended real numbers, and $\delta_\omega$ is the Dirac measure at $\omega$.
From this definition, it is seen that the name '''discrete probability measure''' is compatible with the notion of discrete measure, as $\Omega$ is countable.
{{MissingLinks|to eg. Definition:Discrete Sample Space and connected stuff}}
\end{definition}