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.

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[edit] 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.

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" E ∩ Lx: 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" E ⊆ S. The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.

[edit] 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 π∗(μ). Then there exists a ν-almost everywhere uniquely determined family of probability measures {μx}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;
  • μx "lives on" the fibre π−1(x): for ν-almost all x ∈ X,
\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 E ⊆ Y, taking f to be the indicator function of E,
\mu (E) = \int_{X} \mu_{x} \left( E \cap \pi^{-1}(x) \right) \, \mathrm{d} \nu (x).

[edit] Applications

[edit] 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 : Y → Xi 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}} \, \mathrm{d} (\pi_{1})_{*} (\mu) (x_{1}).

[edit] 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.

[edit] References

  • 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.