Consider the unit square ${\displaystyle S=[0,1]\times [0,1]}$  in the Euclidean plane ${\displaystyle \mathbb {R} ^{2}}$ . Consider the probability measure ${\displaystyle \mu }$  defined on ${\displaystyle S}$  by the restriction of two-dimensional Lebesgue 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 a complete measure space,

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 the zero measure. The probability of a "two-dimensional" event ${\displaystyle E}$  could then be obtained as an integral 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

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

• Let ${\displaystyle Y}$  and ${\displaystyle X}$  be two Radon spaces (i.e. a topological space such that every Borel probability measure on it is inner regular, e.g. separably metrizable spaces; in particular, every probability measure on it is outright a Radon 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 the pushforward measure ${\displaystyle \nu =\pi _{*}(\mu )=\mu \circ \pi ^{-1}}$ . 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" the fiber ${\displaystyle \pi ^{-1}(x)}$ : 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 ${\displaystyle f:Y\to [0,\infty ]}$ ,
  $${\displaystyle \int _{Y}f(y)\,\mathrm {d} \mu (y)=\int _{X}\int _{\pi ^{-1}(x)}f(y)\,\mathrm {d} \mu _{x}(y)\,\mathrm {d} \nu (x).}$$

   
  In particular, for any event ${\displaystyle E\subseteq Y}$ , taking ${\displaystyle f}$  to be the indicator function of ${\displaystyle E}$ ,^[1]
  $${\displaystyle \mu (E)=\int _{X}\mu _{x}(E)\,\mathrm {d} \nu (x).}$$

   

Product spaces edit

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 a Cartesian product ${\displaystyle Y=X_{1}\times X_{2}}$  and ${\displaystyle \pi _{i}:Y\to X_{i}}$  is the natural projection, then each fibre ${\displaystyle \pi _{1}^{-1}(x_{1})}$  can be canonically 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 ${\displaystyle (\pi _{1})_{*}(\mu )}$ -almost everywhere uniquely determined) such that

The relation to conditional expectation is given by the identities

Vector calculus edit

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 surface ${\displaystyle \Sigma \subset \mathbb {R} ^{3}}$ , it is implicit that the "correct" measure on ${\displaystyle \Sigma }$  is the disintegration of three-dimensional Lebesgue measure ${\displaystyle \lambda ^{3}}$  on ${\displaystyle \Sigma }$ , and that the disintegration of this measure on ∂Σ is the same as the disintegration of ${\displaystyle \lambda ^{3}}$  on ${\displaystyle \partial \Sigma }$ .^[2]

Conditional distributions edit

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]

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