In probability theory, the Doob–Dynkin lemma, named after Joseph L. Doob and Eugene Dynkin (also known as the factorization lemma), 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
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.
-algebra of Borel sets on
is a measurable space, then is the smallest
a measurable space.
The "if" part simply states that the composition of two measurable functions is measurable.
The "only if" part is proven below.
First, note that, by the above descriptive definition of
as the set of preimages of
is an indicator of some set
, then the function
suits the requirement, and since
By linearity, the claim extends to any simple measurable function
be measurable but not necessarily simple.
As explained in the article on simple functions,
is a pointwise limit of a monotonically non-decreasing sequence
of simple functions.
The previous step guarantees that
for some measurable
exists on the entire
(The article on measurable functions explains why supremum of a sequence of measurable functions is measurable).
The lemma remains valid if the space
and the bijection is measurable in both directions.
By definition, the measurability of
for every Borel set
and the lemma may be restated as follows.
is a measurable space.