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 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 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,

E\subseteq L_{{x}}\implies \mu (E)=0.

While true, this is somewhat unsatisfying. It would be nice to say that μ "restricted to" L_x 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" E ∩ L_x: more formally, if μ_x denotes one-dimensional Lebesgue measure on L_x, then

\mu (E)=\int _{{[0,1]}}\mu _{{x}}(E\cap L_{{x}})\,{\mathrm {d}}x

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 $\nu$ ∈ P(X) be the pushforward measure $\nu$ = π_∗(μ) = μ ∘ π⁻¹. Then there exists a $\nu$ -almost everywhere uniquely determined family of probability measures {μ_x}_x∈X ⊆ P(Y) 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 ⊆ Y;
μ_x "lives on" the fiber π⁻¹(x): for $\nu$ -almost all x ∈ X,

\mu _{{x}}\left(Y\setminus \pi ^{{-1}}(x)\right)=0,

and so μ_x(E) = μ_x(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 E ⊆ Y, taking f to be the indicator function of E,^[1]

\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 π_i : Y → X_i 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.^[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]

References

↑ Dellacherie, C.; Meyer, P.-A. (1978). Probabilities and potential. North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam.
↑ 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 3-7643-2428-7.
↑ Chang, J.T.; Pollard, D. (1997). "Conditioning as disintegration" (PDF). Statistica Neerlandica. 51 (3): 287. doi:10.1111/1467-9574.00056.

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[Dellacherie_Meyer-1] Dellacherie, C.; Meyer, P.-A. (1978). Probabilities and potential. North-Holland Mathematics Studies, North-Holland Publishing Co., Amsterdam.

[Ambrosio_Gigli_Savare-2] 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 3-7643-2428-7.

[Chang_Pollard-3] Chang, J.T.; Pollard, D. (1997). "Conditioning as disintegration" (PDF). Statistica Neerlandica. 51 (3): 287. doi:10.1111/1467-9574.00056.