Definition:Discrete Probability Measure
Definition
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 $\ds \sum_{\omega \mathop \in \Omega} p_\omega = 1$.
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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.
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Discrete Probability Space
The measure space $\struct {\Omega, \powerset \Omega, P}$ is called discrete probability space.
Also see
Sources
- 2005: René L. Schilling: Measures, Integrals and Martingales ... (previous) ... (next): $4.7 \ \text{(iv)}$