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.
[2] These set families have applications in measure theory and probability.
A major application of 𝜆-systems is the π-𝜆 theorem, see below.
satisfies: Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class.
[note 3] 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.
there exists a unique Dynkin system denoted
is called the Dynkin system generated by
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.
, which is equivalent to showing that the Lebesgue measure is unique on
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
whereas the seemingly more general law of the variable is the probability measure
(on two possibly different probability spaces) are equal in distribution (or law), denoted by
The motivation for the definition stems from the observation that if
A similar result holds for the joint distribution of a random vector.
are two random variables defined on the same probability space
The joint cumulative distribution function of
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
The proof of this is another application of the π-𝜆 theorem.
[4] This article incorporates material from Dynkin system on PlanetMath, which is licensed under the Creative Commons Attribution/Share-Alike License.
Additionally, a semiring is a π-system where every complement
is equal to a finite disjoint union of sets in
is equal to a finite disjoint union of sets in