Doob–Dynkin lemma

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In mathematics, in particular 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 \sigma-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 \sigma-algebra generated by the other.

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

Statement of the lemma[edit]

Let \Omega be a sample space. For a function f:\Omega \rightarrow R^n, the \sigma-algebra generated by f is defined as the family of sets f^{-1}(S), where S are all Borel sets.

Lemma Let X,Y: \Omega \rightarrow R^n be random elements and \sigma(X) be the \sigma algebra generated by X. Then Y is \sigma(X)-measurable if and only if Y=g(X) for some Borel measurable function g:R^n\rightarrow R^n.

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, Y being \sigma(X)-measurable is the same as Y^{-1}(S)\in \sigma(X) for any Borel set S, which is the same as \sigma(Y) \subset \sigma(X). So, the lemma can be rewritten in the following, equivalent form.

Lemma Let X,Y: \Omega \rightarrow R^n be random elements and \sigma(X) and \sigma(Y) the \sigma algebras generated by X and Y, respectively. Then Y=g(X) for some Borel measurable function g:R^n\rightarrow R^n if and only if \sigma(Y) \subset \sigma(X).

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, Band 582, Springer-Verlag (2006), ISBN 0-387-27730-7