# Disintegration theorem

__: Theorem in measure theory__

**Short description**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 **R**^{2}, *S* = [0, 1] × [0, 1]. Consider the probability measure μ defined on *S* by the restriction of two-dimensional Lebesgue measure λ^{2} 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*.

Consider a one-dimensional subset of *S* such as the line segment *L*_{x} = {*x*} × [0, 1]. *L*_{x} has μ-measure zero; every subset of *L*_{x} is a μ-null set; since the Lebesgue measure space is a complete measure space,
[math]\displaystyle{ E \subseteq L_{x} \implies \mu (E) = 0. }[/math]

While true, this is somewhat unsatisfying. It would be nice to say that μ "restricted to" *L*_{x} is the one-dimensional Lebesgue measure λ^{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* ∩ *L*_{x}: more formally, if μ_{x} denotes one-dimensional Lebesgue measure on *L*_{x}, then
[math]\displaystyle{ \mu (E) = \int_{[0, 1]} \mu_{x} (E \cap L_{x}) \, \mathrm{d} x }[/math]
for any "nice" *E* ⊆ *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*M*is inner regular e.g. 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. Here one should think of π as a function to "disintegrate"*Y*, in the sense of partitioning*Y*into [math]\displaystyle{ \{ \pi^{-1}(x)\ |\ x \in X\} }[/math]. For example, for the motivating example above, one can define [math]\displaystyle{ \pi((a,b)) = a, (a,b) \in [0,1]\times [0,1] }[/math], which gives that [math]\displaystyle{ \pi^{-1}(a) = a \times [0,1] }[/math], a slice we want to capture. - Let [math]\displaystyle{ \nu }[/math] ∈
(**P***X*) be the pushforward measure ν = π_{∗}(μ) = μ ∘ π^{−1}. This measure provides the distribution of x (which corresponds to the events [math]\displaystyle{ \pi^{-1}(x) }[/math]).

The conclusion of the theorem: There exists a [math]\displaystyle{ \nu }[/math]-almost everywhere uniquely determined family of probability measures {μ_{x}}_{x∈X} ⊆ * P*(

*Y*), which provides a "disintegration" of [math]\displaystyle{ \mu }[/math] into [math]\displaystyle{ \{\mu_x\}_{x \in X} }[/math], such that:

- the function [math]\displaystyle{ x \mapsto \mu_{x} }[/math] is Borel measurable, in the sense that [math]\displaystyle{ x \mapsto \mu_{x} (B) }[/math] is a Borel-measurable function for each Borel-measurable set
*B*⊆*Y*; - μ
_{x}"lives on" the fiber π^{−1}(*x*): for [math]\displaystyle{ \nu }[/math]-almost all*x*∈*X*, [math]\displaystyle{ \mu_{x} \left( Y \setminus \pi^{-1} (x) \right) = 0, }[/math] and so μ_{x}(*E*) = μ_{x}(*E*∩ π^{−1}(*x*)); - for every Borel-measurable function
*f*:*Y*→ [0, ∞], [math]\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). }[/math] In particular, for any event*E*⊆*Y*, taking*f*to be the indicator function of*E*,^{[1]}[math]\displaystyle{ \mu (E) = \int_{X} \mu_{x} \left( E \right) \, \mathrm{d} \nu (x). }[/math]

## 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} × *X*_{2} and π_{i} : *Y* → *X*_{i} is the natural projection, then each fibre *π*_{1}^{−1}(*x*_{1}) can be canonically identified with *X*_{2} and there exists a Borel family of probability measures [math]\displaystyle{ \{ \mu_{x_{1}} \}_{x_{1} \in X_{1}} }[/math] in * P*(

*X*

_{2}) (which is (π

_{1})

_{∗}(μ)-almost everywhere uniquely determined) such that [math]\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})_{*} (\mu) (x_{1}), }[/math] which is in particular

^{[clarification needed]}[math]\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|x_1) \right) \mu\left( \pi_1^{-1}(\mathrm{d} x_{1})\right) }[/math] and [math]\displaystyle{ \mu(A \times B) = \int_A \mu\left(B|x_1\right) \, \mu\left( \pi_1^{-1}(\mathrm{d} x_{1})\right). }[/math]

The relation to conditional expectation is given by the identities [math]\displaystyle{ \operatorname E(f|\pi_1)(x_1)= \int_{X_2} f(x_1,x_2) \mu(\mathrm d x_2|x_1), }[/math] [math]\displaystyle{ \mu(A\times B|\pi_1)(x_1)= 1_A(x_1) \cdot \mu(B| x_1). }[/math]

### 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 ⊂ **R**^{3}, it is implicit that the "correct" measure on Σ is the disintegration of three-dimensional Lebesgue measure λ^{3} on Σ, and that the disintegration of this measure on ∂Σ is the same as the disintegration of λ^{3} on ∂Σ.^{[2]}

### 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.^{[3]}

## See also

- Ionescu-Tulcea theorem
- Joint probability distribution – Type of probability distribution
- Conditional expectation – Expected value of a random variable given that certain conditions are known to occur
- Regular conditional probability

## References

- ↑ Dellacherie, C.; Meyer, P.-A. (1978).
*Probabilities and Potential*. North-Holland Mathematics Studies. Amsterdam: North-Holland. ISBN 0-7204-0701-X. - ↑ Ambrosio, L., Gigli, N. & Savaré, G. (2005).
*Gradient Flows in Metric Spaces and in the Space of Probability Measures*. ETH Zürich, Birkhäuser Verlag, Basel. ISBN 978-3-7643-2428-5. - ↑ Chang, J.T.; Pollard, D. (1997). "Conditioning as disintegration".
*Statistica Neerlandica***51**(3): 287. doi:10.1111/1467-9574.00056. http://www.stat.yale.edu/~jtc5/papers/ConditioningAsDisintegration.pdf.

Original source: https://en.wikipedia.org/wiki/Disintegration theorem.
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