Inmathematics, thedisintegration theorem is a result inmeasure theory andprobability theory. It rigorously defines the idea of a non-trivial "restriction" of ameasure to ameasure zero subset of themeasure space in question. It is related to the existence ofconditional probability measures. In a sense, "disintegration" is the opposite process to the construction of aproduct measure.

Consider the unit square${\displaystyle S=[0,1]\times [0,1]}$ in theEuclidean plane${\displaystyle {\mathbb{R}}^2}$. Consider theprobability measure${\displaystyle \mu }$ defined on${\displaystyle S}$ by the restriction of two-dimensionalLebesgue measure${\displaystyle {\lambda }^2}$ to${\displaystyle S}$. That is, the probability of an event${\displaystyle E\subseteq S}$ is simply the area of${\displaystyle E}$. We assume${\displaystyle E}$ is a measurable subset of${\displaystyle S}$.

Consider a one-dimensional subset of${\displaystyle S}$ such as the line segment${\displaystyle L_x=\{x\}\times [0,1]}$.${\displaystyle L_x}$ has${\displaystyle \mu }$-measure zero; every subset of${\displaystyle L_x}$ is a${\displaystyle \mu }$-null set; since the Lebesgue measure space is acomplete measure space,$${\displaystyle E\subseteq L_x⟹\mu (E)=0.}$$

While true, this is somewhat unsatisfying. It would be nice to say that${\displaystyle \mu }$ "restricted to"${\displaystyle L_x}$ is the one-dimensional Lebesgue measure${\displaystyle {\lambda }^1}$, rather than thezero measure. The probability of a "two-dimensional" event${\displaystyle E}$ could then be obtained as anintegral of the one-dimensional probabilities of the vertical "slices"${\displaystyle E\cap L_x}$: more formally, if${\displaystyle {\mu }_x}$ denotes one-dimensional Lebesgue measure on${\displaystyle L_x}$, then$${\displaystyle \mu (E)={\int }_{[0,1]}{\mu }_x(E\cap L_x)\mathrm{d}x}$$for any "nice"${\displaystyle E\subseteq S}$. The disintegration theorem makes this argument rigorous in the context of measures onmetric spaces.

(Hereafter,${\displaystyle \mathcal{P}(X)}$ will denote the collection ofBorel probability measures on atopological space${\displaystyle (X,T)}$.)The assumptions of the theorem are as follows:

• Let${\displaystyle Y}$ and${\displaystyle X}$ be twoRadon spaces (i.e. atopological space such that everyBorelprobability measure on it isinner regular, e.g.separably metrizable spaces; in particular, every probability measure on it is outright aRadon measure).
• Let${\displaystyle \mu \in \mathcal{P}(Y)}$.
• Let${\displaystyle \pi :Y\to X}$ be a Borel-measurable function. Here one should think of${\displaystyle \pi }$ as a function to "disintegrate"${\displaystyle Y}$, in the sense of partitioning${\displaystyle Y}$ into${\displaystyle \{{\pi }^{-1}(x)|x\in X\}}$. For example, for the motivating example above, one can define${\displaystyle \pi ((a,b))=a}$,${\displaystyle (a,b)\in [0,1]\times [0,1]}$, which gives that${\displaystyle {\pi }^{-1}(a)=a\times [0,1]}$, a slice we want to capture.
• Let${\displaystyle \nu \in \mathcal{P}(X)}$ be thepushforward measure${\displaystyle \nu ={\pi }_{\ast }(\mu )=\mu \circ {\pi }^{-1}}$. This measure provides the distribution of${\displaystyle x}$ (which corresponds to the events${\displaystyle {\pi }^{-1}(x)}$).

The conclusion of the theorem: There exists a${\displaystyle \nu }$-almost everywhere uniquely determined family of probability measures${\displaystyle \{{\mu }_x{\}}_{x\in X}\subseteq \mathcal{P}(Y)}$, which provides a "disintegration" of${\displaystyle \mu }$ into${\displaystyle \{{\mu }_x{\}}_{x\in X}}$, such that:

• the function${\displaystyle x\mapsto {\mu }_x}$ is Borel measurable, in the sense that${\displaystyle x\mapsto {\mu }_x(B)}$ is a Borel-measurable function for each Borel-measurable set${\displaystyle B\subseteq Y}$;
• ${\displaystyle {\mu }_x}$ "lives on" thefiber${\displaystyle {\pi }^{-1}(x)}$: for${\displaystyle \nu }$-almost all${\displaystyle x\in X}$,$${\displaystyle {\mu }_x\left(Y\setminus {\pi }^{-1}(x)\right)=0,}$$ and so${\displaystyle {\mu }_x(E)={\mu }_x(E\cap {\pi }^{-1}(x))}$;
• for every Borel-measurable function${\displaystyle f:Y\to [0,\mathrm{\infty }]}$,$${\displaystyle {\int }_{Y}f(y)\mathrm{d}\mu (y)={\int }_X{\int }_{{\pi }^{-1}(x)}f(y)\mathrm{d}{\mu }_x(y)\mathrm{d}\nu (x).}$$ In particular, for any event${\displaystyle E\subseteq Y}$, taking${\displaystyle f}$ to be theindicator function of${\displaystyle E}$,^[1]$${\displaystyle \mu (E)={\int }_X{\mu }_x(E)\mathrm{d}\nu (x).}$$

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The original example was a special case of the problem of product spaces, to which the disintegration theorem applies.

When${\displaystyle Y}$ is written as aCartesian product${\displaystyle Y=X_1\times X_2}$ and${\displaystyle {\pi }_i:Y\to X_i}$ is the naturalprojection, then each fibre${\displaystyle {\pi }_1^{-1}(x_1)}$ can becanonically identified with${\displaystyle X_2}$ and there exists a Borel family of probability measures${\displaystyle \{{\mu }_{x_1}{\}}_{x_1\in X_1}}$ in${\displaystyle \mathcal{P}(X_2)}$ (which is${\displaystyle ({\pi }_1{)}_{\ast }(\mu )}$-almost everywhere uniquely determined) such that$${\displaystyle \mu ={\int }_{X_1}{\mu }_{x_1}\mu \left({\pi }_1^{-1}(\mathrm{d}{x}_1)\right)={\int }_{X_1}{\mu }_{x_1}\mathrm{d}({\pi }_1{)}_{\ast }(\mu )(x_1),}$$which is in particular^{[clarification needed]}$${\displaystyle {\int }_{X_1\times X_2}f(x_1,x_2)\mu (\mathrm{d}{x}_1,\mathrm{d}{x}_2)={\int }_{X_1}\left({\int }_{X_2}f(x_1,x_2)\mu (\mathrm{d}{x}_2\mid x_1)\right)\mu \left({\pi }_1^{-1}(\mathrm{d}{x}_1)\right)}$$and$${\displaystyle \mu (A\times B)={\int }_A\mu \left(B\mid x_1\right)\mu \left({\pi }_1^{-1}(\mathrm{d}{x}_1)\right).}$$

The relation toconditional expectation is given by the identities$${\displaystyle \mathrm{E}(f\mid {\pi }_1)(x_1)={\int }_{X_2}f(x_1,x_2)\mu (\mathrm{d}{x}_2\mid x_1),}$$$${\displaystyle \mu (A\times B\mid {\pi }_1)(x_1)=1_A(x_1)\cdot \mu (B\mid x_1).}$$

The disintegration theorem can also be seen as justifying the use of a "restricted" measure invector calculus. For instance, inStokes' theorem as applied to avector field flowing through acompactsurface${\displaystyle \mathrm{\Sigma }\subset {\mathbb{R}}^3}$, it is implicit that the "correct" measure on${\displaystyle \mathrm{\Sigma }}$ is the disintegration of three-dimensional Lebesgue measure${\displaystyle {\lambda }^3}$ on${\displaystyle \mathrm{\Sigma }}$, and that the disintegration of this measure on ∂Σ is the same as the disintegration of${\displaystyle {\lambda }^3}$ on${\displaystyle \mathrm{\partial }\mathrm{\Sigma }}$.^[2]

The disintegration theorem can be applied to give a rigorous treatment of conditional probability distributions in statistics, while avoiding purely abstract formulations of conditional probability.^[3] The theorem is related to theBorel–Kolmogorov paradox, for example.

• Ionescu-Tulcea theorem – Probability theorem
• Joint probability distribution – Type of probability distribution
• Copula (statistics) – Statistical distribution for dependence between random variables
• Conditional expectation – Expected value of a random variable given that certain conditions are known to occur
• Borel–Kolmogorov paradox
• Regular conditional probability

• Theorems in measure theory
• Probability theorems

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