In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin (also known as the factorization lemma), characterizes the situation when one random variable is a function of another by the inclusion of the -algebras generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being measurable with respect to the -algebra generated by the other.

The lemma plays an important role in the conditional expectation in probability theory, where it allows replacement of the conditioning on a random variable by conditioning on the -algebra that is generated by the random variable.

Notations and introductory remarks

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In the lemma below,   is the  -algebra of Borel sets on   If   and   is a measurable space, then

 

is the smallest  -algebra on   such that   is  -measurable.

Statement of the lemma

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Let   be a function, and   a measurable space. A function   is  -measurable if and only if   for some  -measurable  [1]

Remark. The "if" part simply states that the composition of two measurable functions is measurable. The "only if" part is proven below.

Remark. The lemma remains valid if the space   is replaced with   where     is bijective with   and the bijection is measurable in both directions.

By definition, the measurability of   means that   for every Borel set   Therefore   and the lemma may be restated as follows.

Lemma. Let     and   is a measurable space. Then   for some  -measurable   if and only if  .

See also

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References

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  1. ^ Kallenberg, Olav (1997). Foundations of Modern Probability. Springer. p. 7. ISBN 0-387-94957-7.
  • A. Bobrowski: Functional analysis for probability and stochastic processes: an introduction, Cambridge University Press (2005), ISBN 0-521-83166-0
  • M. M. Rao, R. J. Swift : Probability Theory with Applications, Mathematics and Its Applications, vol. 582, Springer-Verlag (2006), ISBN 0-387-27730-7 doi:10.1007/0-387-27731-5