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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 $\\Sigma$.\nAssume that $\\nu$ is [[absolutely continuous|P0P21KDV6E]] with respect to $\\mu$, that is, $\\mu(A) = 0$ implies $\\nu(A) = 0$ for $A \\in \\Sigma$.\nThen there is a [[measurable function|YYNCJSR0PJ]] $f \\colon X \\to [0, \\infty)$ such that\n$$\\nu(A) = \\int_A f \\, d\\mu \\quad \\text{for all } A \\in \\Sigma.$$\nThe function $f$ is unique up to equality $\\mu$-almost everywhere.\n\n## Remarks\n\nA standard proof first treats finite measures.\nConsider the measurable $g \\ge 0$ with $\\int_A g \\, d\\mu \\le \\nu(A)$ for all $A$, and take one that maximises $\\int_X g \\, d\\mu$.\nIf the leftover measure $\\nu - g \\mu$ were not zero, a Hahn decomposition of $\\nu - g\\mu - \\varepsilon \\mu$ would allow $g$ to be increased, a contradiction.\nVon Neumann gave another proof through the Riesz representation of a bounded linear functional on $L^2(\\mu + \\nu)$.\nThe σ-finite case follows by cutting $X$ into countably many pieces of finite measure.\n\nExample: if $\\nu$ is the standard normal distribution on $\\R$ and $\\mu$ is Lebesgue measure, then $f(x) = e^{-x^2/2} / \\sqrt{2\\pi}$.\n\nThe hypothesis of σ-finiteness of $\\mu$ cannot be dropped.\nLet $\\mu$ be counting measure on the Borel sets of $[0, 1]$ and $\\nu$ Lebesgue measure.\nThen $\\nu \\ll \\mu$, because only the empty set has $\\mu$-measure $0$.\nA density $f$ would satisfy $f(x) = \\nu(\\{x\\}) = 0$ for every $x$, which gives $\\nu = 0$, a contradiction.\n\nThe function $f$ is the [[measure with density|Y98C5T956Q]] description of $\\nu$: $\\nu = f \\mu$.\n\n## Explanations\n\n### L1 human\n\nIf one measure never sees mass where another sees none, then the first is the second one weighted by a density function.\nThis is how a probability density arises from a distribution that has no point masses and no mass on sets of length zero.\nThe density plays the role of a derivative of one measure with respect to the other.\n\n### L2 human\n\nThe theorem is the source of the Radon–Nikodym derivative $d\\nu / d\\mu$.\nIn probability it gives conditional expectation: the conditional expectation of an integrable variable given a sub-σ-algebra is a Radon–Nikodym derivative.\nIt also identifies the dual of $L^p(\\mu)$ with $L^q(\\mu)$ for $1 \\le p \u003c \\infty$ and σ-finite $\\mu$.\nTogether with the Lebesgue decomposition it describes every σ-finite measure relative to a fixed one.\n",
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