Disintegration theorem

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 to a measure zero subset of the measure space in question. It is related to the existence of conditional probability measures. In a sense, "disintegration" is the opposite process to the construction of a product measure.

Motivation

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

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

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

Statement of the theorem

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

• Let $Y$ and $X$ 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 $\mu \in 𝒫(Y)$.
• Let $\pi :Y\to X$ be a Borel-measurable function. Here one should think of $\pi$ as a function to "disintegrate" $Y$, in the sense of partitioning $Y$ into $\{{\pi }^{-1}(x)|x\in X\}$. For example, for the motivating example above, one can define $\pi ((a,b))=a$, $(a,b)\in [0,1]\times [0,1]$, which gives that ${\pi }^{-1}(a)=a\times [0,1]$, a slice we want to capture.
• Let $\nu \in 𝒫(X)$ be the pushforward measure $\nu ={\pi }_{*}(\mu )=\mu \circ {\pi }^{-1}$. This measure provides the distribution of $x$ (which corresponds to the events ${\pi }^{-1}(x)$).

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

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

Applications

Product spaces

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

When $Y$ is written as a Cartesian product $Y=X_1\times X_2$ and ${\pi }_i:Y\to X_i$ is the natural projection, then each fibre ${\pi }_1^{-1}(x_1)$ can be canonically identified with $X_2$ and there exists a Borel family of probability measures $\{{\mu }_{x_1}{\}}_{x_1\in X_1}$ in $𝒫(X_2)$ (which is $({\pi }_1{)}_{*}(\mu )$-almost everywhere uniquely determined) such that $$\mu =\underset{X_1}{\int }{\mu }_{x_1}\mu ({\pi }_1^{-1}(\mathrm{d}{x}_1))=\underset{X_1}{\int }{\mu }_{x_1}\mathrm{d}({\pi }_1{)}_{*}(\mu )(x_1),$$ which is in particular $$\underset{X_1\times X_2}{\int }f(x_1,x_2)\mu (\mathrm{d}{x}_1,\mathrm{d}{x}_2)=\underset{X_1}{\int }(\underset{X_2}{\int }f(x_1,x_2)\mu (\mathrm{d}{x}_2\mid x_1))\mu ({\pi }_1^{-1}(\mathrm{d}{x}_1))$$ and $$\mu (A\times B)=\underset{A}{\int }\mu (B\mid x_1)\mu ({\pi }_1^{-1}(\mathrm{d}{x}_1)).$$

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

Vector calculus

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

Conditional distributions

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. The theorem is related to the Borel–Kolmogorov paradox, for example.

See also

• 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 – Conditional probability paradox
• Regular conditional probability
• Lifting theory