In
probability theory
Probability theory or probability calculus 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 expre ...
, the Doob–Dynkin lemma, named after
Joseph L. Doob and
Eugene Dynkin
Eugene Borisovich Dynkin (; 11 May 1924 – 14 November 2014) was a Soviet and American mathematician. He made contributions to the fields of probability and algebra, especially semisimple Lie groups, Lie algebras, and Markov processes. The Dynk ...
(also known as the factorization lemma), characterizes the situation when one
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
is a function of another by the
inclusion
Inclusion or Include may refer to:
Sociology
* Social inclusion, action taken to support people of different backgrounds sharing life together.
** Inclusion (disability rights), promotion of people with disabilities sharing various aspects of lif ...
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 magnitude, mass, and probability of events. These seemingly distinct concepts hav ...
with respect to the
-algebra generated by the other.
The lemma plays an important role in the
conditional expectation
In probability theory, the conditional expectation, conditional expected value, or conditional mean of a random variable is its expected value evaluated with respect to the conditional probability distribution. If the random variable can take on ...
in probability theory, where it allows replacement of the conditioning on a
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a Mathematics, mathematical formalization of a quantity or object which depends on randomness, random events. The term 'random variable' in its mathema ...
by conditioning on the
-algebra that is
generated by the random variable.
Notations and introductory remarks
In the lemma below,