Definition:Discrete Probability Measure

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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