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 (Dynkin himself used this term) or d-system. These set families have applications in measure theory and probability.

A major application of -systems is the - theorem, see below.

Definition

be a collection of subsets is a subset of the power set is a Dynkin system if

1. is closed under complements

2. is closed under countable increasing unions is an increasing sequence

It is easy to check satisfies:

is closed under countable unions of pairwise disjoint is a sequence of pairwise disjoint

• ).

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 -system (that is, closed under finite intersections) 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.

is called the

Sierpiński–Dynkin's π-λ theoremSierpiński-Dynkin's π-λ theoremDynkin's π-λ theoremπ-λ theoremPi-lambda theorem

Sierpiński-Dynkin's - theorem: is a -system

In other words, the 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 (known as the Lebesgue measure):

be the unit interval [0,1] with the Lebesgue measure on Borel sets be another measure is the satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's -.

Application to probability distributions

See also

Notes

References

Further reading