Doob–Dynkin lemma

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 $\sigma$ 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

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

• 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