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, . 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'' = × , 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,
$E \subseteq L_ \implies \mu (E) = 0.$

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
$\mu (E) = \int_ \mu_ (E \cap L_) \, \mathrm x$
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 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 $\$. For example, for the motivating example above, one can define \pi((a,b)) = a, (a,b) \in ,1times ,1/math>, which gives that \pi^(a) = a \times ,1/math>, a slice we want to capture.
* Let $\nu$ ∈ ''P''(''X'') be the pushforward measure This measure provides the distribution of x (which corresponds to the events $\pi^(x)$).

The conclusion of the theorem: There exists a $\nu$- almost everywhere uniquely determined family of probability measures _{''x''∈''X''} ⊆ ''P''(''Y''), which provides a "disintegration" of $\mu$ into such that:
* the function $x \mapsto \mu_$ is Borel measurable, in the sense that $x \mapsto \mu_ (B)$ 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'', $\mu_ \left( Y \setminus \pi^ (x) \right) = 0,$ and so μ_''x''(''E'') = μ_''x''(''E'' ∩ π⁻¹(''x''));
* for every Borel-measurable function ''f'' : ''Y'' → , ∞ $\int_ f(y) \, \mathrm \mu (y) = \int_ \int_ f(y) \, \mathrm \mu_ (y) \mathrm \nu (x).$ In particular, for any event ''E'' ⊆ ''Y'', taking ''f'' to be the indicator function of ''E'', $\mu (E) = \int_ \mu_ \left( E \right) \, \mathrm \nu (x).$

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'' = ''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 (π₁)_∗(μ)-almost everywhere uniquely determined) such that
$\mu = \int_ \mu_ \, \mu \left(\pi_1^(\mathrm d x_1) \right)= \int_ \mu_ \, \mathrm (\pi_)_ (\mu) (x_),$
which is in particular
$\int_ f(x_1,x_2)\, \mu(\mathrm d x_1,\mathrm d x_2) = \int_\left( \int_ f(x_1,x_2) \mu(\mathrm d x_2, x_1) \right) \mu\left( \pi_1^(\mathrm x_)\right)$
and
$\mu(A \times B) = \int_A \mu\left(B, x_1\right) \, \mu\left( \pi_1^(\mathrm x_)\right).$

The relation to conditional expectation is given by the identities
$\operatorname E(f, \pi_1)(x_1)= \int_ f(x_1,x_2) \mu(\mathrm d x_2, x_1),$
$\mu(A\times B, \pi_1)(x_1)= 1_A(x_1) \cdot \mu(B, x_1).$

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 , 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 conditional probability distributions in statistics, while avoiding purely abstract formulations of conditional probability.

See also

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* Regular conditional probability

References

{{Measure theory

Theorems in measure theory
Probability theorems