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. 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 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 has \mu-measure zero; every subset of L_x is a \mu-null set; since the Lebesgue measure space is a complete measure space, 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 for any "nice". The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.

Statement of the theorem

(Hereafter, will denote the collection of Borel probability measures on a topological space (X, T).) The assumptions of the theorem are as follows: The conclusion of the theorem: There exists a \nu-almost everywhere uniquely determined family of probability measures, which provides a "disintegration" of \mu into , such that:

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 and is the natural projection, then each fibre can be canonically identified with X_2 and there exists a Borel family of probability measures in (which is -almost everywhere uniquely determined) such that which is in particular and The relation to conditional expectation is given by the identities

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.

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.