# Disintegration theorem

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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,

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

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 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 :

*Y*→

*X*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 {μ

_{x}}

_{x∈X}⊆

*(*

**P***Y*) such that

- the function is Borel measurable, in the sense that is a Borel-measurable function for each Borel-measurable set
*B*⊆*Y*; - μ
_{x}"lives on" the fiber π^{−1}(*x*): for ν-almost all*x*∈*X*,

- and so μ
_{x}(*E*) = μ_{x}(*E*∩ π^{−1}(*x*));

- for every Borel-measurable function
*f*:*Y*→ [0, ∞],

- In particular, for any event
*E*⊆*Y*, taking*f*to be the indicator function of*E*,^{[1]}

## 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 in * P*(

*X*

_{2}) (which is (π

_{1})

_{∗}(μ)-almost everywhere uniquely determined) such that

which is in particular

and

The relation to conditional expectation is given by the identities

### 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 conditioning probability distributions in statistics, while avoiding purely abstract formulations of conditional probability.^{[3]}

## See also

## References

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