Dynkin system
A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set satisfying a set of axioms weaker than those of -algebra. Dynkin systems are sometimes referred to as -systems or d-system. These set families have applications in measure theory and probability.
A major application of -systems is the - theorem, see below.
Definition
Let be a nonempty set, and let be a collection of subsets of . Then is a Dynkin system if- is closed under complements of subsets in supersets: if and then
- is closed under countable increasing unions: if is an increasing sequence of sets in then
- is closed under complements in : if then
- Taking shows that
- is closed under countable unions of pairwise disjoint sets: if is a sequence of pairwise disjoint sets in then
- To be clear, this property also holds for finite sequences of pairwise disjoint sets.
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 -system is a -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
For instance,
For another example, let and ; then
Sierpiński–Dynkin's π-λ theorem
Sierpiński-Dynkin's - theorem:If is a -system and is a Dynkin system with then
In other words, the -algebra generated by is contained in Thus a Dynkin system contains a -system if and only if it contains the -algebra generated by that -system.
One application of Sierpiński-Dynkin's - theorem is the uniqueness of a measure that evaluates the length of an interval :
Let be the unit interval 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 -algebra generated by It may be shown that satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's - Theorem it follows that in fact includes all of, which is equivalent to showing that the Lebesgue measure is unique on.