# Definition:Discrete Probability Measure

## Definition

Let $\Omega$ be a countable set.

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:

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

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**.

## Also see

## Sources

- 2005: René L. Schilling:
*Measures, Integrals and Martingales*... (previous) ... (next): $4.7 \ \text{(iv)}$