Definition:Discrete Probability Measure

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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 $\ds \sum_{\omega \mathop \in \Omega} p_\omega = 1$.


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.

Discrete Probability Space

The measure space $\struct {\Omega, \powerset \Omega, P}$ is called discrete probability space.


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