# Disintegration theorem

Template:More footnotes 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 in the Euclidean plane R2, S = [0, 1] × [0, 1]. Consider the probability measure μ defined on S by the restriction of two-dimensional Lebesgue measure λ2 to S. That is, the probability of an event ES 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 Lx = {x} × [0, 1]. Lx has μ-measure zero; every subset of Lx is a μ-null set; since the Lebesgue measure space is a complete measure space,

$E\subseteq L_{x}\implies \mu (E)=0.$ While true, this is somewhat unsatisfying. It would be nice to say that μ "restricted to" Lx is the one-dimensional Lebesgue measure λ1, 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" ELx: more formally, if μx denotes one-dimensional Lebesgue measure on Lx, then

$\mu (E)=\int _{[0,1]}\mu _{x}(E\cap L_{x})\,{\mathrm {d} }x$ for any "nice" ES. 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 metric space (X, d).)

Let Y and X be two Radon spaces (i.e. separable metric spaces on which every probability measure is a Radon measure). Let μ ∈ P(Y), let π : YX be a Borel-measurable function, and let ν ∈ P(X) be the pushforward measure ν = π(μ) = μ ∘ π−1. Then there exists a ν-almost everywhere uniquely determined family of probability measures {μx}xXP(Y) such that

$\mu _{x}\left(Y\setminus \pi ^{-1}(x)\right)=0,$ and so μx(E) = μx(E ∩ π−1(x));
• for every Borel-measurable function f : Y → [0, ∞],
$\int _{Y}f(y)\,{\mathrm {d} }\mu (y)=\int _{X}\int _{\pi ^{-1}(x)}f(y)\,{\mathrm {d} }\mu _{x}(y){\mathrm {d} }\nu (x).$ In particular, for any event EY, taking f to be the indicator function of E,
$\mu (E)=\int _{X}\mu _{x}\left(E\right)\,{\mathrm {d} }\nu (x).$ ## 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 = X1 × X2 and πi : YXi is the natural projection, then each fibre π1−1(x1) can be canonically identified with X2 and there exists a Borel family of probability measures $\{\mu _{x_{1}}\}_{x_{1}\in X_{1}}$ in P(X2) (which is (π1)(μ)-almost everywhere uniquely determined) such that

$\mu =\int _{X_{1}}\mu _{x_{1}}\,\mu \left(\pi _{1}^{-1}({\mathrm {d} }x_{1})\right)=\int _{X_{1}}\mu _{x_{1}}\,{\mathrm {d} }(\pi _{1})_{*}(\mu )(x_{1}),$ which is in particular

$\int _{X_{1}\times X_{2}}f(x_{1},x_{2})\,\mu ({\mathrm {d} }x_{1},{\mathrm {d} }x_{2})=\int _{X_{1}}\left(\int _{X_{2}}f(x_{1},x_{2})\mu ({\mathrm {d} }x_{2}|x_{1})\right)\mu \left(\pi _{1}^{-1}({\mathrm {d} }x_{1})\right)$ and

$\mu (A\times B)=\int _{A}\mu \left(B|x_{1}\right)\,\mu \left(\pi _{1}^{-1}({\mathrm {d} }x_{1})\right).$ The relation to conditional expectation is given by the identities

$\operatorname {E} (f|\pi _{1})(x_{1})=\int _{X_{2}}f(x_{1},x_{2})\mu ({\mathrm {d} }x_{2}|x_{1}),$ $\mu (A\times B|\pi _{1})(x_{1})=1_{A}(x_{1})\cdot \mu (B|x_{1}).$ ### 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 Σ ⊂ R3, it is implicit that the "correct" measure on Σ is the disintegration of three-dimensional Lebesgue measure λ3 on Σ, and that the disintegration of this measure on ∂Σ is the same as the disintegration of λ3 on ∂Σ.

### Conditional distributions

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.