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  "body": "\n## Statement\n\nLet $(X, \\Sigma)$ be a measurable space and let $\\mu$ and $\\nu$ be [[σ-finite measures|WEBBT18D4X]] on $\\Sigma$.\nThen there are measures $\\nu_a$ and $\\nu_s$ on $\\Sigma$ such that:\n\n1. $\\nu = \\nu_a + \\nu_s$.\n2. $\\nu_a$ is [[absolutely continuous|P0P21KDV6E]] with respect to $\\mu$.\n3. $\\nu_s$ and $\\mu$ are [[mutually singular|EBGECZNCV5]].\n\nThe measures $\\nu_a$ and $\\nu_s$ are unique.\n\n## Remarks\n\nOne proof applies the Radon–Nikodym theorem to $\\mu$ and $\\mu + \\nu$.\nWith $f = d\\mu / d(\\mu + \\nu)$, the set $S = \\{f = 0\\}$ is $\\mu$-null, and one puts $\\nu_s(A) = \\nu(A \\cap S)$ and $\\nu_a(A) = \\nu(A \\setminus S)$.\nUniqueness holds because a measure that is both absolutely continuous and singular with respect to $\\mu$ is zero.\n\nExample: let $\\mu$ be Lebesgue measure on $\\R$ and $\\nu = \\mu|_{[0,1]} + \\delta_2$, where $\\delta_2$ is the point mass at $2$.\nThen $\\nu_a = \\mu|_{[0,1]}$ and $\\nu_s = \\delta_2$.\n\nOn the real line with $\\mu$ Lebesgue measure, the singular part splits further into a part made of point masses and a continuous singular part.\nThe distribution of the Cantor function is an example of a continuous singular measure.\n\n## Explanations\n\n### L1 human\n\nAny reasonable measure can be cut into two pieces relative to a reference measure.\nOne piece is smooth: it has a density with respect to the reference.\nThe other piece lives on a set that the reference does not see at all.\n\n### L2 human\n\nCombined with the Radon–Nikodym theorem, it says $\\nu = f\\mu + \\nu_s$ with $f = d\\nu_a / d\\mu$ and $\\nu_s \\perp \\mu$.\nIn probability this splits a distribution on $\\R$ into a part with a density, a discrete part and a singular continuous part.\nThe same decomposition holds for finite signed and complex measures, applied to the total variation.\n",
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