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. Dynkin systems are sometimes referred to as 𝜆-systems (Dynkin himself used this term) or d-system.[2] These set families have applications in measure theory and probability.

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


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

  1. is closed under complements of subsets in supersets: if and then
  2. is closed under countable increasing unions: if is an increasing sequence[note 1] of sets in then

It is easy to check[proof 1] that any Dynkin system satisfies:

  1. is closed under complements in : if then
    • Taking shows that
  2. is closed under countable unions of pairwise disjoint sets: if is a sequence of pairwise disjoint sets in (meaning that for all ) then
    • To be clear, this property also holds for finite sequences of pairwise disjoint sets (by letting for all ).

Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class.[proof 2] 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.

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 Dynkin system generated by For instance, For another example, let and ; then

Sierpiński–Dynkin's π-λ theorem

Sierpiński-Dynkin's π-𝜆 theorem:[3] If is a π-system and is a Dynkin system with then

In other words, the 𝜎-algebra generated by is contained in 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 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 𝜎-algebra generated by It may be shown that satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's π-𝜆 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

This section is transcluded from pi system. (edit | history)

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
where is the Borel 𝜎-algebra. The random variables and (on two possibly different probability spaces) are equal in distribution (or law), denoted by if they have the same cumulative distribution functions; that is, if The motivation for the definition stems from the observation that if then that is exactly to say that and agree on the π-system which generates and so by the example above:

A similar result holds for the joint distribution of a random vector. For example, suppose and are two random variables defined on the same probability space with respectively generated π-systems and The joint cumulative distribution function of is

However, and Because

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 In other words, and have the same distribution if and only if they have the same joint cumulative distribution function.

In the theory of stochastic processes, two processes are known to be equal in distribution if and only if they agree on all finite-dimensional distributions; that is, for all

The proof of this is another application of the π-𝜆 theorem.[4]

See also


  1. ^ A sequence of sets is called increasing if for all


  1. ^ Assume satisfies (1), (2), and (3). Proof of (5) :Property (5) follows from (1) and (2) by using The following lemma will be used to prove (6). Lemma: If are disjoint then Proof of Lemma: implies where by (5). Now (2) implies that contains so that (5) guarantees that which proves the lemma. Proof of (6) Assume that are pairwise disjoint sets in For every integer the lemma implies that where because is increasing, (3) guarantees that contains their union as desired.
  2. ^ Assume satisfies (4), (5), and (6). proof of (2): If satisfy then (5) implies and since (6) implies that contains so that finally (4) guarantees that is in Proof of (3): Assume is an increasing sequence of subsets in let and let for every where (2) guarantees that all belong to Since are pairwise disjoint, (6) guarantees that their union belongs to which proves (3).
  1. ^ Dynkin, E., "Foundations of the Theory of Markov Processes", Moscow, 1959
  2. ^ Aliprantis, Charalambos; Border, Kim C. (2006). Infinite Dimensional Analysis: a Hitchhiker's Guide (Third ed.). Springer. Retrieved August 23, 2010.
  3. ^ Sengupta. "Lectures on measure theory lecture 6: The Dynkin π − λ Theorem" (PDF). Math.lsu. Retrieved 3 January 2023.
  4. ^ Kallenberg, Foundations Of Modern Probability, p. 48


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