Jordan decomposition (of a measure)

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For the Jordan decomposition of a signed measure we refer to Jordan decomposition (of a signed measure).

In probability theory, the Jordan decomposition of a probability measure $\mu$ is given as $\mu = p \mu_a + (1-p)\mu_{na}$, where $p\in [0,1]$, $\mu_a$ is an atomic probability measure and $\mu_{na}$ is a nonatomic probability measure. The decomposition into atomic and nonatomic part holds in general for $\sigma$-finite measures. See also Atom.