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.

Notations and introductory remarks

In the lemma below, ${\displaystyle {\overline {\mathbb {R} }}=\mathbb {R} \cup \{-\infty \}\cup \{+\infty \}}$ is the extended real number line, and ${\displaystyle {\mathcal {B}}({\overline {\mathbb {R} }})}$ is the ${\displaystyle \sigma }$-algebra of Borel sets on ${\displaystyle {\overline {\mathbb {R} }}.}$ The notation ${\displaystyle g:(X,{\mathcal {X}})\rightarrow (Y,{\mathcal {Y}})}$ indicates that ${\displaystyle g}$ is a function from ${\displaystyle X}$ to ${\displaystyle Y,}$ and that ${\displaystyle g}$ is measurable relative to the ${\displaystyle \sigma }$-algebras ${\displaystyle {\mathcal {X}}}$ and ${\displaystyle {\mathcal {Y}}.}$

Furthermore, if ${\displaystyle T:X\to Y,}$ and ${\displaystyle (Y,{\mathcal {Y}})}$ is a measurable space, we define

${\displaystyle \sigma (T)=\{T^{-1}(S)\mid S\in {\mathcal {Y}}\}.}$

One can easily check that ${\displaystyle \sigma (T)}$ is the minimal ${\displaystyle \sigma }$-algebra on ${\displaystyle X}$ under which ${\displaystyle T}$ is measurable, i.e.

${\displaystyle T:(X,\sigma (T))\to (Y,{\mathcal {Y}}).}$

Statement of the lemma

Let ${\displaystyle T:\Omega \rightarrow \Omega '}$ be a function from a set ${\displaystyle \Omega }$ to a measurable space ${\displaystyle (\Omega ',{\mathcal {A}}'),}$ and ${\displaystyle \operatorname {Im} T}$ is ${\displaystyle {\mathcal {A}}'}$-measurable. Further, let ${\displaystyle f:\Omega \rightarrow {\overline {\mathbb {R} }}}$ be a scalar function on ${\displaystyle \Omega }$. Then ${\displaystyle f}$ is ${\displaystyle \sigma (T)}$-measurable if and only if ${\displaystyle f=g\circ T,}$ for some measurable function ${\displaystyle g:(\Omega ',{\mathcal {A}}')\rightarrow ({\overline {\mathbb {R} }},{\mathcal {B}}({\overline {\mathbb {R} }})).}$

Note. The "if" part simply states that the composition of two measurable functions is measurable. The "only if" part is proven below.

Proof.

Let ${\displaystyle f}$ be ${\displaystyle \sigma (T)}$-measurable. Step 1: assume that ${\displaystyle f}$ is a simple function, i.e. ${\displaystyle \textstyle f=\sum _{i=1}^{n}f_{i}\,\cdot {\mathbf {1} }_{A_{i}},}$ for some non-empty pairwise disjoint sets ${\displaystyle \{A_{i}\}_{i=1}^{n}}$ from ${\displaystyle \sigma (T).}$ If ${\displaystyle A_{i}=T^{-1}(A_{i}'),}$ then the function ${\displaystyle g=\textstyle \sum _{i=1}^{n}f_{i}\,\cdot {\mathbf {1} }_{A_{i}'}}$ suits the requirement. Step 2: if ${\displaystyle f\geq 0}$, then ${\displaystyle f}$ is a pointwise limit of a non-decreasing sequence ${\displaystyle f_{n}}$ of simple functions (see the article on simple functions for the proof). Step 1 guarantees that ${\displaystyle f_{n}=g_{n}\circ T.}$ This equality, in turn, implies that the sequence ${\displaystyle g_{n}(x)}$ is non-decreasing, as long as ${\displaystyle x\in \operatorname {Im} T,}$ so the function ${\displaystyle {\widetilde {g}}(x)=\lim _{n\to \infty }g_{n}(x)}$ is well defined (finite or infinite) for every ${\displaystyle x\in \operatorname {Im} T.}$ As a pointwise limit of measurable ${\displaystyle {\overline {\mathbb {R} }}}$-valued functions, ${\displaystyle {\widetilde {g}}}$ is itself measurable (see the article on measurable functions). Define ${\displaystyle g(x)={\begin{cases}{\widetilde {g}}(x)&\quad {\text{if }}x\in \operatorname {Im} T\\[1mm]0&\quad {\text{if }}x\notin \operatorname {Im} T.\end{cases}}}$ The measurability of ${\displaystyle g}$ is based on the assumption that ${\displaystyle \operatorname {Im} T\in {\mathcal {A}}'.}$ Thus, ${\displaystyle g}$ suits the requirement. Step 3: every measurable function ${\displaystyle f}$ is the difference of its positive and negative parts, i.e. ${\displaystyle f=f^{+}-f^{-},}$ where both ${\displaystyle f^{+}}$ and ${\displaystyle f^{-}}$ are measurable and non-negative. Step 2 guarantees that ${\displaystyle f^{+}=g^{+}\circ T}$ and ${\displaystyle f^{-}=g^{-}\circ T.}$ Define ${\displaystyle g(x)={\begin{cases}g^{+}(x)-g^{-}(x)&\quad {\text{if }}x\in \operatorname {Im} T\\[1mm]0&\quad {\text{if }}x\notin \operatorname {Im} T.\end{cases}}}$ Since it is not possible that ${\displaystyle f^{+}(x)>0}$ and ${\displaystyle f^{-}(x)>0}$ for the same ${\displaystyle x,}$ the equality ${\displaystyle g^{+}(x)=g^{-}(x)=\infty }$ never holds, and hence ${\displaystyle g}$ is well defined. Being the difference of two measurable functions, ${\displaystyle g^{+}-g^{-}}$ is also measurable. Since ${\displaystyle \operatorname {Im} T}$ is measurable, so is ${\displaystyle g.}$ Thus, ${\displaystyle g}$ suits the requirement.

By definition, ${\displaystyle f}$ being ${\displaystyle \sigma (T)}$-measurable is the same as ${\displaystyle f^{-1}(S)\in \sigma (T)}$ for every Borel set ${\displaystyle S}$, which is the same as ${\displaystyle \sigma (f)\subseteq \sigma (T)}$. So, the lemma can be rewritten in the following, equivalent form.

Lemma. Let ${\displaystyle f}$ and ${\displaystyle T}$ be as above. Then ${\displaystyle f=g\circ T,}$ for some Borel function ${\displaystyle g,}$ if and only if ${\displaystyle \sigma (f)\subseteq \sigma (T)}$.

See also

• Conditional expectation

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