Doob–Dynkin lemma
Encyclopedia
In mathematics
, the Doob–Dynkin lemma, named after Joseph Doob and Eugene Dynkin
, is a statement in probability theory
that characterizes the situation when one random variable
is a function of another, in terms of measurability and
algebras.
and 
be the 
algebra generated by 
. Then 
is 
measurable if and only if 
for some Borel measurable function 
.
Mathematics
Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...
, the Doob–Dynkin lemma, named after Joseph Doob and Eugene Dynkin
Eugene Dynkin
Eugene Borisovich Dynkin is a Soviet and American mathematician. He has made contributions to the fields of probability and algebra, especially semisimple Lie groups, Lie algebras, and Markov processes...
, is a statement in probability theory
Probability theory
Probability theory is the branch of mathematics concerned with analysis of random phenomena. The central objects of probability theory are random variables, stochastic processes, and events: mathematical abstractions of non-deterministic events or measured quantities that may either be single...
that characterizes the situation when one random variable
Random variable
In probability and statistics, a random variable or stochastic variable is, roughly speaking, a variable whose value results from a measurement on some type of random process. Formally, it is a function from a probability space, typically to the real numbers, which is measurable functionmeasurable...
is a function of another, in terms of measurability and
algebras.Statement of the lemma
Let and be the algebra generated by . Then is measurable if and only if for some Borel measurable function .