Family closed under complements and countable disjoint unions
A Dynkin system,[1] named after Eugene Dynkin, is a collection of subsets of another universal set
satisfying a set of axioms weaker than those of 𝜎-algebra. Dynkin systems are sometimes referred to as 𝜆-systems (Dynkin himself used this term) or d-system.[2] These set families have applications in measure theory and probability.
A major application of 𝜆-systems is the π-𝜆 theorem, see below.
Definition
Let
be a nonempty set, and let
be a collection of subsets of
(that is,
is a subset of the power set of
). Then
is a Dynkin system if
![{\displaystyle \Omega \in D;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/1591a4abb374c33f5124647dbbcf6fba1b26c6cc)
is closed under complements of subsets in supersets: if
and
then ![{\displaystyle B\setminus A\in D;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8c4bf351aa176243ca7aad146307b8a91e2f409d)
is closed under countable increasing unions: if
is an increasing sequence[note 1] of sets in
then ![{\displaystyle \bigcup _{n=1}^{\infty }A_{n}\in D.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/416f0ae523bdee76eedebbd5b5ca4ae321a466ba)
It is easy to check[proof 1] that any Dynkin system
satisfies:
![{\displaystyle \varnothing \in D;}](https://wikimedia.org/api/rest_v1/media/math/render/svg/67c2cfbe657acfdd208870d2081cd57aa7f49f04)
is closed under complements in
: if
then
- Taking
shows that ![{\displaystyle \varnothing \in D.}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d68e8bb42a80f6ed80008ea723c80840e357801)
is closed under countable unions of pairwise disjoint sets: if
is a sequence of pairwise disjoint sets in
(meaning that
for all
) then
- To be clear, this property also holds for finite sequences
of pairwise disjoint sets (by letting
for all
).
Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class.[proof 2]
For this reason, a small group of authors have adopted conditions 4-6 to define a Dynkin system.
An important fact is that any Dynkin system that is also a π-system (that is, closed under finite intersections) is a 𝜎-algebra. This can be verified by noting that conditions 2 and 3 together with closure under finite intersections imply closure under finite unions, which in turn implies closure under countable unions.
Given any collection
of subsets of
there exists a unique Dynkin system denoted
which is minimal with respect to containing
That is, if
is any Dynkin system containing
then
is called the Dynkin system generated by
For instance,
For another example, let
and
; then
Sierpiński–Dynkin's π-λ theorem
Sierpiński-Dynkin's π-𝜆 theorem:[3]
If
is a π-system and
is a Dynkin system with
then
In other words, the 𝜎-algebra generated by
is contained in
Thus a Dynkin system contains a π-system if and only if it contains the 𝜎-algebra generated by that π-system.
One application of Sierpiński-Dynkin's π-𝜆 theorem is the uniqueness of a measure that evaluates the length of an interval (known as the Lebesgue measure):
Let
be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let
be another measure on
satisfying
and let
be the family of sets
such that
Let
and observe that
is closed under finite intersections, that
and that
is the 𝜎-algebra generated by
It may be shown that
satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's π-𝜆 Theorem it follows that
in fact includes all of
, which is equivalent to showing that the Lebesgue measure is unique on
.
Application to probability distributions
The π-𝜆 theorem motivates the common definition of the probability distribution of a random variable
in terms of its cumulative distribution function. Recall that the cumulative distribution of a random variable is defined as
![{\displaystyle F_{X}(a)=\operatorname {P} [X\leq a],\qquad a\in \mathbb {R} ,}](https://wikimedia.org/api/rest_v1/media/math/render/svg/ce7f397fb0c38c13d07c65d9fcf82497904cef89)
whereas the seemingly more general law of the variable is the probability measure
![{\displaystyle {\mathcal {L))_{X}(B)=\operatorname {P} \left[X^{-1}(B)\right]\quad {\text{ for all ))B\in {\mathcal {B))(\mathbb {R} ),}](https://wikimedia.org/api/rest_v1/media/math/render/svg/37a28bbc802a4b857d4f7433d2d3861de4de4c51)
where
is the Borel 𝜎-algebra. The random variables
and
(on two possibly different probability spaces) are equal in distribution (or law), denoted by
if they have the same cumulative distribution functions; that is, if
The motivation for the definition stems from the observation that if
then that is exactly to say that
and
agree on the π-system
which generates
and so by the example above:
A similar result holds for the joint distribution of a random vector. For example, suppose
and
are two random variables defined on the same probability space
with respectively generated π-systems
and
The joint cumulative distribution function of
is
![{\displaystyle F_{X,Y}(a,b)=\operatorname {P} [X\leq a,Y\leq b]=\operatorname {P} \left[X^{-1}((-\infty ,a])\cap Y^{-1}((-\infty ,b])\right],\quad {\text{ for all ))a,b\in \mathbb {R} .}](https://wikimedia.org/api/rest_v1/media/math/render/svg/e7a2b7fac21544d1da36d39beccfa789d7138083)
However,
and
Because
![{\displaystyle {\mathcal {I))_{X,Y}=\left\{A\cap B:A\in {\mathcal {I))_{X},{\text{ and ))B\in {\mathcal {I))_{Y}\right\))](https://wikimedia.org/api/rest_v1/media/math/render/svg/fd007253f5d59913b6998701931638536b994f54)
is a π-system generated by the random pair
the π-𝜆 theorem is used to show that the joint cumulative distribution function suffices to determine the joint law of
In other words,
and
have the same distribution if and only if they have the same joint cumulative distribution function.
In the theory of stochastic processes, two processes
are known to be equal in distribution if and only if they agree on all finite-dimensional distributions; that is, for all
![{\displaystyle \left(X_{t_{1)),\ldots ,X_{t_{n))\right)\,{\stackrel {\mathcal {D)){=))\,\left(Y_{t_{1)),\ldots ,Y_{t_{n))\right).}](https://wikimedia.org/api/rest_v1/media/math/render/svg/d39e3036e32b1a78feac8c75720f3b178606fc99)
The proof of this is another application of the π-𝜆 theorem.[4]