A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set \Omega 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. 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 \Omega be a nonempty set, and let D be a collection of subsets of \Omega (that is, D is a subset of the power set of \Omega). Then D is a Dynkin system if It is easy to check that any Dynkin system D satisfies:

<li></li> <li>D is closed under complements in \Omega: if A \in D, then <li>D is closed under countable unions of [pairwise disjoint](https://bliptext.com/articles/pairwise-disjoint) sets: if is a sequence of [pairwise disjoint](https://bliptext.com/articles/pairwise-disjoint) sets in D (meaning that for all i \neq j) then </ol> Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class. 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](https://bliptext.com/articles/pi-system) (that is, closed under finite intersections) is a [𝜎-algebra](https://bliptext.com/articles/sigma-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 \mathcal{J} of subsets of \Omega, there exists a unique Dynkin system denoted which is minimal with respect to containing \mathcal J. That is, if \tilde D is any Dynkin system containing then is called the For instance, For another example, let and ; then

Sierpiński–Dynkin's π-λ theorem

Sierpiński-Dynkin's π-𝜆 theorem: If P is a π-system and D is a Dynkin system with then In other words, the 𝜎-algebra generated by P is contained in D. 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 m be another measure on \Omega satisfying and let D be the family of sets S such that Let and observe that I is closed under finite intersections, that and that \mathcal{B} is the 𝜎-algebra generated by I. It may be shown that D satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's π-𝜆 Theorem it follows that D in fact includes all of \mathcal{B}, which is equivalent to showing that the Lebesgue measure is unique on \mathcal{B}.

Application to probability distributions