Lambda system

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 -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

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
# $\Omega \in D;$
# $D$ is closed under complements of subsets in supersets: if $A, B \in D$ and $A \subseteq B,$ then $B \setminus A \in D;$
# $D$ is closed under countable increasing unions: if $A_1 \subseteq A_2 \subseteq A_3 \subseteq \cdots$ is an increasing sequenceA sequence of sets $A_1, A_2, A_3, \ldots$ is called if $A_n \subseteq A_$ for all $n \geq 1.$ of sets in $D$ then $\bigcup_^\infty A_n \in D.$

It is easy to check that any Dynkin system $D$ satisfies:

4. $\varnothing \in D;$


5. $D$ is closed under complements in $\Omega$: if $A \in D,$ then $\Omega \setminus A \in D;$
   * Taking $A := \Omega$ shows that $\varnothing \in D.$


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 $A_i \cap A_j = \varnothing$ for all $i \neq j$) then $\bigcup_^\infty A_n \in D.$
   * 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 $\mathcal$ of subsets of $\Omega,$ there exists a unique Dynkin system denoted $D\$ which is minimal with respect to containing $\mathcal J.$ That is, if $\tilde D$ is any Dynkin system containing $\mathcal,$ then $D\ \subseteq \tilde.$ $D\$ is called the
For instance, $D\ = \.$
For another example, let $\Omega = \$ and $\mathcal = \$; then $D\ = \.$

Sierpiński–Dynkin's π-λ theorem

Sierpiński-Dynkin's - theorem:
If $P$ is a -system and $D$ is a Dynkin system with $P\subseteq D,$ then $\sigma\\subseteq D.$

In other words, the -algebra generated by $P$ is contained in $D.$ 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 $(\Omega, \mathcal, \ell)$ be the unit interval ,1with the Lebesgue measure on Borel sets. Let $m$ be another measure on $\Omega$ satisfying m a, b)= b - a, and let $D$ be the family of sets $S$ such that m = \ell Let $I := \,$ and observe that $I$ is closed under finite intersections, that $I \subseteq D,$ and that $\mathcal$ is the -algebra generated by $I.$ It may be shown that $D$ satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's - Theorem it follows that $D$ in fact includes all of $\mathcal$, which is equivalent to showing that the Lebesgue measure is unique on $\mathcal$.

Application to probability distributions

See also

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Notes

References

Further reading

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{{Measure theory

Families of sets
Lemmas
Probability theory