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 σ -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 σ -algebra generated by the other.
The lemma plays an important role in the conditional expectation in probability theory, where it allows to replace the conditioning on a random variable by conditioning on the σ -algebra that is generated by the random variable.
Let Ω be a sample space. For a function f : Ω → R n , the σ -algebra generated by f is defined as the family of sets f − 1 ( S ) , where S are all Borel sets.
Lemma Let X , Y : Ω → R n be random elements and σ ( X ) be the σ algebra generated by X . Then Y is σ ( X ) -measurable if and only if Y = g ( X ) for some Borel measurable function g : R n → 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 σ ( X ) -measurable is the same as Y − 1 ( S ) ∈ σ ( X ) for any Borel set S , which is the same as σ ( Y ) ⊂ σ ( X ) . So, the lemma can be rewritten in the following, equivalent form.
Lemma Let X , Y : Ω → R n be random elements and σ ( X ) and σ ( Y ) the σ algebras generated by X and Y , respectively. Then Y = g ( X ) for some Borel measurable function g : R n → R n if and only if σ ( Y ) ⊂ σ ( X ) .
