Statement in probability theory
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
-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
-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,
is the
-algebra of Borel sets on
If
and
is a measurable space, then
![{\displaystyle \sigma (T)\ {\stackrel {\text{def}}{=}}\ \{T^{-1}(S)\mid S\in {\mathcal {Y}}\}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8e53829c815fd1ea111c3143093a5eb4fdd7b1f5)
is the smallest
-algebra on
such that
is
-measurable.
Statement of the lemma
Let
be a function, and
a measurable space. A function
is
-measurable if and only if
for some
-measurable
[1]
Remark. The "if" part simply states that the composition of two measurable functions is measurable. The "only if" part is proven below.
Proof. |
Let be -measurable. Assume that is an indicator of some set If then the function suits the requirement. By linearity, the claim extends to any simple measurable function Let be measurable but not necessarily simple. As explained in the article on simple functions, is a pointwise limit of a monotonically non-decreasing sequence of simple functions. The previous step guarantees that for some measurable The supremum exists on the entire and is measurable. (The article on measurable functions explains why supremum of a sequence of measurable functions is measurable). For every the sequence is non-decreasing, so which shows that |
Remark. The lemma remains valid if the space
is replaced with
where
is bijective with
and the bijection is measurable in both directions.
By definition, the measurability of
means that
for every Borel set
Therefore
and the lemma may be restated as follows.
Lemma. Let
and
is a measurable space. Then
for some
-measurable
if and only if
.
See also
References
- ^ Kallenberg, Olav (1997). Foundations of Modern Probability. Springer. p. 7. ISBN 0-387-94957-7.
- A. Bobrowski: Functional analysis for probability and stochastic processes: an introduction, Cambridge University Press (2005), ISBN 0-521-83166-0
- M. M. Rao, R. J. Swift : Probability Theory with Applications, Mathematics and Its Applications, vol. 582, Springer-Verlag (2006), ISBN 0-387-27730-7 doi:10.1007/0-387-27731-5