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

\sigma

-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

\sigma

-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

by conditioning on the

\sigma

-algebra that is generated by the random variable.

Notations and introductory remarks

In the lemma below,

\mathcal,1 /math> is the \sigma -algebra of

Borel set In mathematics, a Borel set is any subset of a topological space that can be formed from its open sets (or, equivalently, from closed sets) through the operations of countable union, countable intersection, and relative complement. Borel sets ...

s on

,1

T\colon X\to Y,

and

(Y,)

is a measurable space, then :

\sigma(T)\ \stackrel\ \

is the smallest

\sigma

-algebra on

X

such that

T

\sigma(T) /

-measurable.

Statement of the lemma

Let

T\colon \Omega\rightarrow\Omega'

be a function, and

(\Omega',\mathcal')

a measurable space. A function

f\colon \Omega\rightarrow,1

\sigma(T) / \mathcal,1

-measurable if and only if

f=g\circ T,

for some

\mathcal' / \mathcal,1

-measurable

g\colon \Omega' \to,1

Remark. The "if" part simply states that the composition of two measurable functions is measurable. The "only if" part is proven below. Remark. The lemma remains valid if the space

(,1 \mathcal,1

is replaced with

(S,\mathcal(S)),

where

S \subseteq \infty,\infty

S

is bijective with

,1

and the bijection is measurable in both directions. By definition, the measurability of

f

means that

f^(S)\in \sigma(T)

for every Borel set

S \subseteq,1

Therefore

\sigma(f) \subseteq \sigma(T),

and the lemma may be restated as follows. Lemma. Let

T\colon \Omega\rightarrow\Omega',

f\colon \Omega\rightarrow,1

and

(\Omega',\mathcal')

is a measurable space. Then

f = g\circ T,

for some

\mathcal' / \mathcal,1

-measurable

g\colon \Omega' \to,1

if and only if

\sigma(f) \subseteq \sigma(T)

References

* A. Bobrowski: ''Functional analysis for probability and stochastic processes: an introduction'', Cambridge University Press (2005), * M. M. Rao, R. J. Swift : ''Probability Theory with Applications'', Mathematics and Its Applications, vol. 582, Springer-Verlag (2006), {{DEFAULTSORT:Doob-Dynkin Lemma Theorems in probability theory Theorems in measure theory

Notations and introductory remarks

Statement of the lemma

See also

References