In
probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expressing it through a set ...
, the Doob–Dynkin lemma, named after
Joseph L. Doob and
Eugene Dynkin
Eugene Borisovich Dynkin (russian: link=no, Евгений Борисович Дынкин; 11 May 1924 – 14 November 2014) was a Soviet and American mathematician. He made contributions to the fields of probability and algebra, especially sem ...
(also known as the factorization lemma), characterizes the situation when one
random variable is a function of another by the
inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, aims to create an environment that supports equal opportunity for individuals and groups that form a society.
** Inclusion (disability rights), promotion of people with disabiliti ...
of the
-algebras generated by the random variables. The usual statement of the lemma is formulated in terms of one random variable being
measurable
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as mass and probability of events. These seemingly distinct concepts have many simila ...
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,