Disintegration theorem on arvau

Wikipedia / 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

in the Euclidean plane. Consider the probability measure by the restriction of two-dimensional Lebesgue measure.

-null set; since the Lebesgue measure space is a complete measure space

, rather than the zero measure could then be obtained as an integral. The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.

Statement of the theorem

will denote the collection of Borel probability measures on a topological space.) The assumptions of the theorem are as follows:

be two Polish spaces (i.e. separably completely metrizable spaces).
.
be a Borel-measurable function, a slice we want to capture.
be the pushforward measure).

-almost everywhere into such that:

;
"lives on" the fiber-almost all;
to be the indicator function, is a regular conditional probability.

Applications

Product spaces

The original example was a special case of the problem of product spaces, to which the disintegration theorem applies.

is written as a Cartesian product is the natural projection can be canonically which is in particular

The relation to conditional expectation

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 .

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.