Doob–Dynkin lemma

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 ${\displaystyle \sigma }$-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.

"Dynkin's lemma" redirects here. For the statement about ${\displaystyle \lambda }$-systems, see Dynkin system § Sierpiński–Dynkin's π-λ theorem.

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.