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 -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 -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 -algebra that is generated by the random variable.

Notations and introductory remarks

In the lemma below, is the extended real number line, and is the -algebra of Borel sets on
The notation indicates that is a function from to and that is measurable relative to the -algebras and
Furthermore, if and is a measurable space, we define
One can easily check that is the minimal -algebra on under which is measurable, i.e.

Statement of the lemma

Let be a function from a set to a measurable space and is -measurable. Further, let be a scalar function on. Then is -measurable if and only if for some measurable function
Note. The "if" part simply states that the composition of two measurable functions is measurable. The "only if" part is proven below.
Proof.

Let be -measurable.
Step 1: assume that is a simple function, i.e. for some non-empty pairwise disjoint sets from If then the function suits the requirement.
Step 2: if , then is a pointwise limit of a non-decreasing sequence of simple functions. Step 1 guarantees that This equality, in turn, implies that the sequence is non-decreasing, as long as so the function
is well defined for every As a pointwise limit of measurable -valued functions, is itself measurable. Define
The measurability of is based on the assumption that Thus, suits the requirement.
Step 3: every measurable function is the difference of its positive and negative parts, i.e. where both and are measurable and non-negative. Step 2 guarantees that and Define
Since it is not possible that and for the same the equality never holds, and hence is well defined.
Being the difference of two measurable functions, is also measurable. Since is measurable, so is Thus, suits the requirement.

By definition, being -measurable is the same as for every Borel set, which is the same as. So, the lemma can be rewritten in the following, equivalent form.
Lemma. Let and be as above. Then for some Borel function if and only if.