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| {{More footnotes|date=December 2010}}
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| In [[mathematics]], the '''disintegration theorem''' is a result in [[measure theory]] and [[probability theory]]. It rigorously defines the idea of a non-trivial "restriction" of a [[measure (mathematics)|measure]] to a [[measure zero]] subset of the [[measure space]] in question. It is related to the existence of [[conditioning (probability)|conditional probability measures]]. In a sense, "disintegration" is the opposite process to the construction of a [[product measure]].
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| ==Motivation==
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| Consider the unit square in the [[Euclidean plane]] '''R'''<sup>2</sup>, ''S'' = [0, 1] × [0, 1]. Consider the [[probability measure]] μ defined on ''S'' by the restriction of two-dimensional [[Lebesgue measure]] λ<sup>2</sup> to ''S''. That is, the probability of an event ''E'' ⊆ ''S'' is simply the area of ''E''. We assume ''E'' is a measurable subset of ''S''.
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| Consider a one-dimensional subset of ''S'' such as the line segment ''L''<sub>''x''</sub> = {''x''} × [0, 1]. ''L''<sub>''x''</sub> has μ-measure zero; every subset of ''L''<sub>''x''</sub> is a μ-[[null set]]; since the Lebesgue measure space is a [[complete measure|complete measure space]],
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| :<math>E \subseteq L_{x} \implies \mu (E) = 0.</math>
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| While true, this is somewhat unsatisfying. It would be nice to say that μ "restricted to" ''L''<sub>''x''</sub> is the one-dimensional Lebesgue measure λ<sup>1</sup>, rather than the [[trivial measure|zero measure]]. The probability of a "two-dimensional" event ''E'' could then be obtained as an [[Lebesgue integration|integral]] of the one-dimensional probabilities of the vertical "slices" ''E'' ∩ ''L''<sub>''x''</sub>: more formally, if μ<sub>''x''</sub> denotes one-dimensional Lebesgue measure on ''L''<sub>''x''</sub>, then
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| :<math>\mu (E) = \int_{[0, 1]} \mu_{x} (E \cap L_{x}) \, \mathrm{d} x</math>
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| for any "nice" ''E'' ⊆ ''S''. The disintegration theorem makes this argument rigorous in the context of measures on [[metric space]]s.
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| ==Statement of the theorem==
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| (Hereafter, '''''P'''''(''X'') will denote the collection of [[Borel measure|Borel]] probability measures on a [[metric space]] (''X'', ''d'').)
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| Let ''Y'' and ''X'' be two [[Radon space]]s (i.e. [[separable space|separable]] metric spaces on which every probability measure is a [[Radon measure]]). Let μ ∈ '''''P'''''(''Y''), let π : ''Y'' → ''X'' be a Borel-[[measurable function]], and let ν ∈ '''''P'''''(''X'') be the [[pushforward measure]] ν = π<sub>∗</sub>(μ) = μ ∘ π<sup>−1</sup>. Then there exists a ν-[[almost everywhere]] uniquely determined family of probability measures {μ<sub>''x''</sub>}<sub>''x''∈''X''</sub> ⊆ '''''P'''''(''Y'') such that
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| * the function <math>x \mapsto \mu_{x}</math> is Borel measurable, in the sense that <math>x \mapsto \mu_{x} (B)</math> is a Borel-measurable function for each Borel-measurable set ''B'' ⊆ ''Y'';
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| * μ<sub>''x''</sub> "lives on" the [[fiber (mathematics)|fiber]] π<sup>−1</sup>(''x''): for ν-[[almost all]] ''x'' ∈ ''X'',
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| ::<math>\mu_{x} \left( Y \setminus \pi^{-1} (x) \right) = 0,</math>
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| :and so μ<sub>''x''</sub>(''E'') = μ<sub>''x''</sub>(''E'' ∩ π<sup>−1</sup>(''x''));
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| * for every Borel-measurable function ''f'' : ''Y'' → [0, ∞],
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| ::<math>\int_{Y} f(y) \, \mathrm{d} \mu (y) = \int_{X} \int_{\pi^{-1} (x)} f(y) \, \mathrm{d} \mu_{x} (y) \mathrm{d} \nu (x).</math>
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| :In particular, for any event ''E'' ⊆ ''Y'', taking ''f'' to be the [[indicator function]] of ''E'',<ref name=Dellacherie_Meyer>{{cite book | author=Dellacherie, C. & Meyer, P.-A. | title=Probabilities and potential| publisher=North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam | year=1978}}</ref>
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| ::<math>\mu (E) = \int_{X} \mu_{x} \left( E \right) \, \mathrm{d} \nu (x).</math>
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| ==Applications==
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| ===Product spaces===
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| The original example was a special case of the problem of product spaces, to which the disintegration theorem applies.
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| When ''Y'' is written as a [[Cartesian product]] ''Y'' = ''X''<sub>1</sub> × ''X''<sub>2</sub> and π<sub>''i''</sub> : ''Y'' → ''X''<sub>''i''</sub> is the natural [[projection (mathematics)|projection]], then each fibre ''π''<sub>1</sub><sup>−1</sup>(''x''<sub>1</sub>) can be [[canonical form|canonically]] identified with ''X''<sub>2</sub> and there exists a Borel family of probability measures <math>\{ \mu_{x_{1}} \}_{x_{1} \in X_{1}}</math> in '''''P'''''(''X''<sub>2</sub>) (which is (π<sub>1</sub>)<sub>∗</sub>(μ)-almost everywhere uniquely determined) such that
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| :<math>\mu = \int_{X_{1}} \mu_{x_{1}} \, \mu \left(\pi_1^{-1}(\mathrm d x_1) \right)= \int_{X_{1}} \mu_{x_{1}} \, \mathrm{d} (\pi_{1})_{*} (\mu) (x_{1}),</math> | |
| which is in particular
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| :<math>\int_{X_1\times X_2} f(x_1,x_2)\, \mu(\mathrm d x_1,\mathrm d x_2) = \int_{X_1}\left( \int_{X_2} f(x_1,x_2) \mu(\mathrm d x_2|x_1) \right) \mu\left( \pi_1^{-1}(\mathrm{d} x_{1})\right)</math>
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| and
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| :<math>\mu(A \times B) = \int_A \mu\left(B|x_1\right) \, \mu\left( \pi_1^{-1}(\mathrm{d} x_{1})\right).</math>
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| The relation to [[conditional expectation]] is given by the identities
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| :<math>\operatorname E(f|\pi_1)(x_1)= \int_{X_2} f(x_1,x_2) \mu(\mathrm d x_2|x_1),</math>
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| :<math>\mu(A\times B|\pi_1)(x_1)= 1_A(x_1) \cdot \mu(B| x_1).</math>
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| ===Vector calculus===
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| The disintegration theorem can also be seen as justifying the use of a "restricted" measure in [[vector calculus]]. For instance, in [[Stokes' theorem]] as applied to a [[vector field]] flowing through a [[compact space|compact]] [[surface]] Σ ⊂ '''R'''<sup>3</sup>, it is implicit that the "correct" measure on Σ is the disintegration of three-dimensional Lebesgue measure λ<sup>3</sup> on Σ, and that the disintegration of this measure on ∂Σ is the same as the disintegration of λ<sup>3</sup> on ∂Σ.<ref name=Ambrosio_Gigli_Savare>{{cite book | author=Ambrosio, L., Gigli, N. & Savaré, G. | title=Gradient Flows in Metric Spaces and in the Space of Probability Measures | publisher=ETH Zürich, Birkhäuser Verlag, Basel | year=2005 | isbn=3-7643-2428-7 }}</ref>
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| ===Conditional distributions===
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| The disintegration theorem can be applied to give a rigorous treatment of conditioning probability distributions in statistics, while avoiding purely abstract formulations of conditional probability.<ref name=Chang_Pollard>{{cite journal|last=Chang|first=J.T.|coauthors=Pollard, D.|title=Conditioning as disintegration|journal=Statistica Neerlandica| year=1997 | volume=51|issue=3|url=http://www.stat.yale.edu/~jtc5/papers/ConditioningAsDisintegration.pdf|doi=10.1111/1467-9574.00056|pages=287}}</ref>
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| ==See also==
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| *[[Joint probability distribution]]
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| *[[Copula (statistics)]]
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| *[[Conditional expectation]]
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| ==References==
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| {{Reflist}}
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| {{Use dmy dates|date=December 2010}}
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| [[Category:Theorems in measure theory]]
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| [[Category:Probability theorems]]
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