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. Consider the probability measure defined on by the restriction of two-dimensional Lebesgue measure to. That is, the probability of an event is simply the area of. We assume is a measurable subset of.
Consider a one-dimensional subset of such as the line segment. has -measure zero; every subset of is a -null set; since the Lebesgue measure space is a [complete measure|complete measure space],
While true, this is somewhat unsatisfying. It would be nice to say that "restricted to" is the one-dimensional Lebesgue measure, rather than the zero measure. The probability of a "two-dimensional" event could then be obtained as an integral of the one-dimensional probabilities of the vertical "slices" : more formally, if denotes one-dimensional Lebesgue measure on, then
for any "nice". The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.

Statement of the theorem

The assumptions of the theorem are as follows:
  • Let and be two Radon spaces.
  • Let.
  • Let be a Borel-measurable function. Here one should think of as a function to "disintegrate", in the sense of partitioning into. For example, for the motivating example above, one can define,, which gives that, a slice we want to capture.
  • Let be the pushforward measure. This measure provides the distribution of .
The conclusion of the theorem: There exists a -almost everywhere uniquely determined family of probability measures, which provides a "disintegration" of into such that:
  • the function is Borel measurable, in the sense that is a Borel-measurable function for each Borel-measurable set ;
  • "lives on" the fiber : for -almost all, and so ;
  • for every Borel-measurable function, In particular, for any event, taking to be the indicator function of,

Applications

Product spaces

The original example was a special case of the problem of product spaces, to which the disintegration theorem applies.
When is written as a Cartesian product and is the natural projection, then each fibre can be canonically identified with and there exists a Borel family of probability measures in 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 is the disintegration of three-dimensional Lebesgue measure on, and that the disintegration of this measure on ∂Σ is the same as the disintegration of 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.