Definition:Discrete Probability Measure

Definition
Let $\Omega$ be a countable set, and let $\mathcal P \left({\Omega}\right)$ be its power set, regarded as a $\sigma$-algebra.

Let $\left({p_\omega}\right)_{\omega \in \Omega} \subseteq \left[{0 .. 1}\right]$ be a subset of the closed unit interval in $\R$, indexed by $\Omega$.

Suppose that $\displaystyle \sum_{\omega \mathop \in \Omega} p_\omega = 1$.

The discrete probability measure on $\Omega$, denoted $P$, is the mapping defined by:


 * $P: \mathcal P \left({\Omega}\right) \to \overline{\R}, \ P \left({S}\right) = \displaystyle \sum_{\omega \mathop \in \Omega} p_\omega \delta_\omega \left({S}\right)$

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.

Discrete Probability Space
The measure space $\left({\Omega, \mathcal P \left({\Omega}\right), P}\right)$ is called discrete probability space.