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 $\sigma$ -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 replacement of 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 $f:\Omega \rightarrow \mathbf {R} ^{n}$ , the $\sigma$ -algebra generated by $f$ is defined as the family of sets $f^{-1}(S)$ , where $S$ are all Borel sets.

Lemma Let $X,Y:\Omega \rightarrow \mathbf {R} ^{n}$ 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 $g:\mathbf {R} ^{n}\rightarrow \mathbf {R} ^{n}$ .

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, $Y$ 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 $X,Y:\Omega \rightarrow \mathbf {R} ^{n}$ 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 $g:\mathbf {R} ^{n}\rightarrow \mathbf {R} ^{n}$ if and only if $\sigma (Y)\subset \sigma (X)$ .