Dynkin system explained

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.^[2] 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 (that is, is a subset of the power set 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^[3] of sets in then

It is easy to check that any Dynkin system

satisfies:

4. \varnothing\inD;

5. D

   is closed under complements in

   \Omega

   : if $A\; \backslash in\; D,$ then

   \Omega\setminusA\inD;

   • Taking

   A:=\Omega

   shows that

   \varnothing\inD.

6. D

   is closed under countable unions of pairwise disjoint sets: if

   A_1,A_2,A_3,\ldots

   is a sequence of pairwise disjoint sets in

   D

   (meaning that

   A_i\capA_j=\varnothing

   for all

   i ≠ j

   ) then

   	infty
   cup
   	n=1

   A_n\inD.

   • To be clear, this property also holds for finite sequences

   A_1,\ldots,A_n

   of pairwise disjoint sets (by letting

   A_i:=\varnothing

   for all

   i>n

   ).

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.

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:^[4] 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 (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 -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 .

Application to probability distributions

Notes

Proofs

References

• Book: Gut , Allan . Probability: A Graduate Course . Springer Texts in Statistics . Springer . 2005 . New York . 10.1007/b138932 . 0-387-22833-0.
• Book: Billingsley , Patrick . Patrick Billingsley . Probability and Measure . John Wiley & Sons, Inc. . 1995 . New York . 0-471-00710-2.
• Book: Williams, David . David Williams (mathematician) . 2007 . Probability with Martingales . Cambridge University Press . 978-0-521-40605-5 . 193 .

Notes and References

1. Dynkin, E., "Foundations of the Theory of Markov Processes", Moscow, 1959
2. Book: Infinite Dimensional Analysis: a Hitchhiker's Guide. Third. Charalambos. Aliprantis. Kim C.. Border . Springer. 2006. 978-3-540-29587-7 . August 23, 2010.
3. A sequence of sets

   A_1,A_2,A_3,\ldots

   is called if

   A_n\subseteqA_n+1

   for all

   n\geq1.

4. Web site: Sengupta . Lectures on measure theory lecture 6: The Dynkin π − λ Theorem . Math.lsu . 3 January 2023.