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Disintegration theorem

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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

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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 for any "nice" . The disintegration theorem makes this argument rigorous in the context of measures on metric spaces.

Statement of the theorem

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

  • Let and be two Radon spaces (i.e. a topological space such that every Borel probability measure on it is inner regular, e.g. separably metrizable spaces; in particular, every probability measure on it is outright a Radon measure).
  • 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 (which corresponds to the events ).

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 ,[1]

Applications

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Product spaces

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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 which is in particular[clarification needed] and

The relation to conditional expectation is given by the identities

Vector calculus

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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]

Conditional distributions

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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.

See also

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References

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  1. ^ Dellacherie, C.; Meyer, P.-A. (1978). Probabilities and Potential. North-Holland Mathematics Studies. Amsterdam: North-Holland. ISBN 0-7204-0701-X.
  2. ^ Ambrosio, L.; Gigli, N.; Savaré, G. (2005). Gradient Flows in Metric Spaces and in the Space of Probability Measures. ETH Zürich, Birkhäuser Verlag, Basel. ISBN 978-3-7643-2428-5.
  3. ^ Chang, J.T.; Pollard, D. (1997). "Conditioning as disintegration" (PDF). Statistica Neerlandica. 51 (3): 287. CiteSeerX 10.1.1.55.7544. doi:10.1111/1467-9574.00056. S2CID 16749932.