Doob–Dynkin lemma

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In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin, 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.

Statement of the lemma[edit]

Let be a sample space. For a function , the -algebra generated by is defined as the family of sets , where are all Borel sets.

Lemma Let be random elements and be the algebra generated by . Then is -measurable if and only if for some Borel measurable function .

The "if" part of the lemma is simply the statement that the composition of two measurable functions is measurable. The "only if" part is the nontrivial one.

By definition, being -measurable is the same as for any Borel set , which is the same as . So, the lemma can be rewritten in the following, equivalent form.

Lemma Let be random elements and and the algebras generated by and , respectively. Then for some Borel measurable function if and only if .

References[edit]

  • 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