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 S = ,1times ,1/math> in the Euclidean plane $\mathbb^2$. Consider the probability measure $\mu$ defined on $S$ by the restriction of two-dimensional Lebesgue measure $\lambda^2$ to $S$. That is, the probability of an event $E\subseteq S$ 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 $L_x = \\times$, 1/math>. $L_x$ has $\mu$-measure zero; every subset of $L_x$ is a $\mu$-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 $\mu$ "restricted to" $L_x$ is the one-dimensional Lebesgue measure $\lambda^1$, 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" $E\cap L_x$: more formally, if $\mu_x$ denotes one-dimensional Lebesgue measure on $L_x$, then
$\mu (E) = \int_ \mu_ (E \cap L_) \, \mathrm x$
for any "nice" $E\subseteq S$. The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.

Statement of the theorem

(Hereafter, $\mathcal(X)$ 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 it is inner regular, e.g. separably metrizable spaces; in particular, every probability measure on it is outright a Radon measure).
* Let $\mu\in\mathcal(Y)$.
* Let $\pi : Y\to X$ be a Borel-measurable function. Here one should think of $\pi$ 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 $\nu \in\mathcal(X)$ be the pushforward measure $\nu = \pi_(\mu) = \mu \circ \pi^$. 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 $\_ \subseteq \mathcal(Y)$, 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 $B\subseteq Y$;
* $\mu_x$ "lives on" the fiber $\pi^(x)$: for $\nu$-almost all $x\in X$, $\mu_ \left( Y \setminus \pi^ (x) \right) = 0,$ and so $\mu_x(E) =\mu_x(E\cap\pi^(x))$;
* for every Borel-measurable function f : Y \to ,\infty/math>, $\int_ f(y) \, \mathrm \mu (y) = \int_ \int_ f(y) \, \mathrm \mu_x (y) \, \mathrm \nu (x).$ In particular, for any event $E\subseteq Y$, taking $f$ to be the indicator function of $E$, $\mu (E) = \int_X \mu_x (E) \, \mathrm \nu (x).$

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 $Y = X_1\times X_2$ and $\pi_i : Y\to X_i$ is the natural projection, then each fibre $\pi_1^(x_1)$ can be canonically identified with $X_2$ and there exists a Borel family of probability measures $\_$ in $\mathcal(X_2)$ (which is $(\pi_1)_*(\mu)$-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
$\int_ f(x_1,x_2)\, \mu(\mathrm d x_1,\mathrm d x_2) = \int_\left( \int_ f(x_1,x_2) \mu(\mathrm d x_2\mid x_1) \right) \mu\left( \pi_1^(\mathrm x_)\right)$
and
$\mu(A \times B) = \int_A \mu\left(B\mid x_1\right) \, \mu\left( \pi_1^(\mathrm x_)\right).$

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

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 $\Sigma$ is the disintegration of three-dimensional Lebesgue measure $\lambda^3$ on $\Sigma$, and that the disintegration of this measure on ∂Σ is the same as the disintegration of $\lambda^3$ on $\partial\Sigma$.

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. The theorem is related to the Borel–Kolmogorov paradox, for example.

See also

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* Regular conditional probability

References

{{Measure theory

Theorems in measure theory
Theorems in probability theory