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  "body": "\n## Statement\n\nLet $Y$ and $X$ be [[Polish spaces|584GXXY73M]] with their Borel σ-algebras.\nLet $\\mu$ be a Borel [[probability measure|059TJN0NZ1]] on $Y$, let $\\pi \\colon Y \\to X$ be a Borel [[measurable map|YYNCJSR0PJ]], and let $\\nu = \\pi_* \\mu$ be the [[pushforward measure|3N66TM3TSQ]].\nThen there is a family $(\\mu_x)_{x \\in X}$ of Borel probability measures on $Y$ with the following properties.\n\n1. For every Borel set $B \\subseteq Y$, the map $x \\mapsto \\mu_x(B)$ is Borel measurable.\n2. For $\\nu$-almost every $x \\in X$, $\\mu_x$ is concentrated on the fibre: $\\mu_x(Y \\setminus \\pi^{-1}(x)) = 0$.\n3. For every Borel function $f \\colon Y \\to [0, \\infty]$,\n$$\\int_Y f \\, d\\mu = \\int_X \\int_{\\pi^{-1}(x)} f(y) \\, d\\mu_x(y) \\, d\\nu(x).$$\n\nThe family is unique up to $\\nu$-null sets: two such families agree for $\\nu$-almost every $x$.\n\n## Remarks\n\nOne way to prove it is through [[regular conditional probabilities|PHHT5W3BEJ]].\nFor a Borel set $B$, the measure $A \\mapsto \\mu(B \\cap \\pi^{-1}(A))$ is absolutely continuous with respect to $\\nu$, and the [[Radon–Nikodym theorem|1ZSZS5G1Q4]] gives a density $x \\mapsto \\mu_x(B)$.\nThe Polish assumption lets one choose these densities for a countable generating algebra of Borel sets at once and extend them to measures.\n\nExample: let $\\mu$ be Lebesgue measure on $Y = [0, 1]^2$ and $\\pi(x, y) = x$.\nThen $\\nu$ is Lebesgue measure on $[0, 1]$, and $\\mu_x$ is Lebesgue measure on the vertical segment $\\set{x} \\times [0, 1]$.\nProperty 3 is then Fubini's theorem.\n\nThe theorem is often stated for Radon spaces, which include Polish spaces.\nWithout such a regularity assumption a disintegration may fail to exist.\n\n## Explanations\n\n### L1 human\n\nA measure on a big space can be cut into slices along the fibres of a map.\nEach slice carries its own probability measure, and the total measure is recovered by averaging the slices according to how much weight each fibre gets.\nIt is the measure-theoretic form of computing a probability by first conditioning on the value of $\\pi$.\n\n### L2 human\n\nDisintegration makes conditioning on events of probability zero rigorous: $\\mu_x$ is the conditional law of the point $y$ given $\\pi(y) = x$.\nIt is used in optimal transport to decompose transport plans, in ergodic theory for the ergodic decomposition, and in geometric measure theory together with the coarea formula.\nThe measures $\\mu_x$ are only defined for almost every $x$.\nSo a conditional law given a single null event depends on the choice of the map $\\pi$, as the [[Borel–Kolmogorov paradox|9RMS0ZJ618]] shows.\n",
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