Disintegration theorem explained

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 = [ 0 , 1 ] x [ 0 , 1 ] in the Euclidean plane R ^{2} . Consider the probability measure \mu defined on S by the restriction of two-dimensional Lebesgue measure λ ^{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} = \{x\} x [ 0 , 1 ] . 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,$$E\; \backslash subseteq\; L\_\; \backslash implies\; \backslash mu\; (E)\; =\; 0.$$

While true, this is somewhat unsatisfying. It would be nice to say that

\mu "restricted to" L _{x} is the one-dimensional Lebesgue measure 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\cap L _{x} : more formally, if \mu _{x} denotes one-dimensional Lebesgue measure on L _{x} , then$$\backslash mu\; (E)\; =\; \backslash int\_\; \backslash mu\_\; (E\; \backslash cap\; L\_)\; \backslash ,\; \backslash mathrm\; x$$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,

l{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 it is inner regular, e.g. separably metrizable spaces; in particular, every probability measure on it is outright a Radon measure).

• Let

\mu\in l{P}(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 \{ \pi ^{-1} (x) | x \in X\} . For example, for the motivating example above, one can define \pi((a,b)) = a , (a,b) \in [ 0 , 1 ] x [ 0 , 1 ] , which gives that \pi ^{-1} (a) = a x [ 0 , 1 ] , a slice we want to capture.

• Let

\nu \in l{P}(X) be the pushforward measure \nu = \pi _{*} (\mu) = \mu \circ \pi ^{-1} . This measure provides the distribution of x (which corresponds to the events \pi ^{-1} (x) ).

The conclusion of the theorem: There exists a

\nu -almost everywhere uniquely determined family of probability measures \{\mu _{x\}} _{x\in} \subseteq l{P}(Y) , which provides a "disintegration" of \mu into 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\subseteq Y ;\mu _{x} "lives on" the fiber \pi ^{-1} (x) : for \nu -almost all x\in X , $$\backslash mu\_\; \backslash left(Y\; \backslash setminus\; \backslash pi^\; (x)\; \backslash right)\; =\; 0,$$ and so \mu _{x(E)} = \mu ^{-1} _{x(E\cap\pi} (x)) ;

• for every Borel-measurable function

f : Y \to [ 0 , inf ty] , $$\backslash int\_\; f(y)\; \backslash ,\; \backslash mathrm\; \backslash mu\; (y)\; =\; \backslash int\_\; \backslash int\_\; f(y)\; \backslash ,\; \backslash mathrm\; \backslash mu\_x\; (y)\; \backslash ,\; \backslash mathrm\; \backslash nu\; (x).$$ In particular, for any event E\subseteq Y , taking f to be the indicator function of E ,^[1] $$\backslash mu\; (E)\; =\; \backslash int\_X\; \backslash mu\_x\; (E)\; \backslash ,\; \backslash mathrm\; \backslash nu\; (x).$$

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} X _{2} and \pi _{i} : Y\to X _{i} is the natural projection, then each fibre \pi ^{-1} _{1} (x _{1)} can be canonically identified with X _{2} and there exists a Borel family of probability measures \{ \mu _{x _{1}} \} _{x _{1} \in X _{1}} in l{P}(X _{2)} (which is (\pi _{1)} _{*(\mu)} -almost everywhere uniquely determined) such that$$\backslash mu\; =\; \backslash int\_\; \backslash mu\_\; \backslash ,\; \backslash mu\; \backslash left(\backslash pi\_1^(\backslash mathrm\; d\; x\_1)\; \backslash right)=\; \backslash int\_\; \backslash mu\_\; \backslash ,\; \backslash mathrm\; (\backslash pi\_)\_\; (\backslash mu)\; (x\_),$$which is in particular$$\backslash int\_\; f(x\_1,x\_2)\backslash ,\; \backslash mu(\backslash mathrm\; d\; x\_1,\backslash mathrm\; d\; x\_2)\; =\; \backslash int\_\backslash left(\backslash int\_\; f(x\_1,x\_2)\; \backslash mu(\backslash mathrm\; d\; x\_2\backslash mid\; x\_1)\; \backslash right)\; \backslash mu\backslash left(\backslash pi\_1^(\backslash mathrm\; x\_)\backslash right)$$and$$\backslash mu(A\; \backslash times\; B)\; =\; \backslash int\_A\; \backslash mu\backslash left(B\backslash mid\; x\_1\backslash right)\; \backslash ,\; \backslash mu\backslash left(\backslash pi\_1^(\backslash mathrm\; x\_)\backslash right).$$

The relation to conditional expectation is given by the identities$$\backslash operatorname\; E(f\backslash mid\; \backslash pi\_1)(x\_1)=\; \backslash int\_\; f(x\_1,x\_2)\; \backslash mu(\backslash mathrm\; d\; x\_2\backslash mid\; x\_1),$$$$\backslash mu(A\backslash times\; B\backslash mid\; \backslash pi\_1)(x\_1)=\; 1\_A(x\_1)\; \backslash cdot\; \backslash mu(B\backslash mid\; x\_1).$$

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 λ ^{3} on \Sigma , and that the disintegration of this measure on ∂Σ is the same as the disintegration of λ ^{3} on \partial\Sigma .^[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] The theorem is related to the Borel–Kolmogorov paradox, for example.

See also

• • • • • • Regular conditional probability

Notes and References

1. Book: Dellacherie, C. . Meyer, P.-A. . Probabilities and Potential . North-Holland Mathematics Studies . North-Holland . Amsterdam . 1978 . 0-7204-0701-X .
2. Book: Ambrosio, L. . Gigli, N. . Savaré, G. . Gradient Flows in Metric Spaces and in the Space of Probability Measures . ETH Zürich, Birkhäuser Verlag, Basel . 2005 . 978-3-7643-2428-5 .
3. Chang . J.T. . Pollard, D. . Conditioning as disintegration . Statistica Neerlandica . 1997 . 51 . 3 . 10.1111/1467-9574.00056 . 287 . 10.1.1.55.7544 . 16749932 .