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.
Consider the unit square
by the restriction of two-dimensional Lebesgue measure
That is, the probability of an event
is simply the area of
-null set; since the Lebesgue measure space is a complete measure space,
While true, this is somewhat unsatisfying.
is the one-dimensional Lebesgue measure
The probability of a "two-dimensional" event
could then be obtained as an integral of the one-dimensional probabilities of the vertical "slices"
denotes one-dimensional Lebesgue measure on
The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.
will denote the collection of Borel probability measures on a topological space
The assumptions of the theorem are as follows: The conclusion of the theorem: There exists a
-almost everywhere uniquely determined family of probability measures
, such that: 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, then each fibre
can be canonically identified with
and there exists a Borel family of probability measures
The relation to conditional expectation is given by the identities
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
[2] 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.