# 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 ${\displaystyle \sigma }$-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 ${\displaystyle \sigma }$-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 ${\displaystyle \sigma }$-algebra that is generated by the random variable.

## Statement of the lemma

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

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

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

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

## 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, vol. 582, Springer-Verlag (2006), ISBN 0-387-27730-7 doi:10.1007/0-387-27731-5