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 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 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
(Hereafter, 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 , which provides a "disintegration" of into , such that:
This section needs additional citations for
verification. (May 2022) |
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 (which is -almost everywhere uniquely determined) such that
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 on , and that the disintegration of this measure on ∂Σ is the same as the disintegration of on . [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]
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 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 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
(Hereafter, 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 , which provides a "disintegration" of into , such that:
This section needs additional citations for
verification. (May 2022) |
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 (which is -almost everywhere uniquely determined) such that
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 on , and that the disintegration of this measure on ∂Σ is the same as the disintegration of on . [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]