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

edit

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

edit

(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

edit

Product spaces

edit

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

edit

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

edit

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

edit

References

edit
  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.