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
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 .
- 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