Dynkin system
A Dynkin system,[1] named after Eugene Dynkin, is a collection of subsets of another universal set [math]\displaystyle{ \Omega }[/math] 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.
Definition
Let [math]\displaystyle{ \Omega }[/math] be a nonempty set, and let [math]\displaystyle{ D }[/math] be a collection of subsets of [math]\displaystyle{ \Omega }[/math] (that is, [math]\displaystyle{ D }[/math] is a subset of the power set of [math]\displaystyle{ \Omega }[/math]). Then [math]\displaystyle{ D }[/math] is a Dynkin system if
- [math]\displaystyle{ \Omega \in D; }[/math]
- [math]\displaystyle{ D }[/math] is closed under complements of subsets in supersets: if [math]\displaystyle{ A, B \in D }[/math] and [math]\displaystyle{ A \subseteq B, }[/math] then [math]\displaystyle{ B \setminus A \in D; }[/math]
- [math]\displaystyle{ D }[/math] is closed under countable increasing unions: if [math]\displaystyle{ A_1 \subseteq A_2 \subseteq A_3 \subseteq \cdots }[/math] is an increasing sequence[note 1] of sets in [math]\displaystyle{ D }[/math] then [math]\displaystyle{ \bigcup_{n=1}^\infty A_n \in D. }[/math]
It is easy to check[proof 1] that any Dynkin system [math]\displaystyle{ D }[/math] satisfies:
- [math]\displaystyle{ \varnothing \in D; }[/math]
- [math]\displaystyle{ D }[/math] is closed under complements in [math]\displaystyle{ \Omega }[/math]: if [math]\displaystyle{ A \in D, }[/math] then [math]\displaystyle{ \Omega \setminus A \in D; }[/math]
- Taking [math]\displaystyle{ A := \Omega }[/math] shows that [math]\displaystyle{ \varnothing \in D. }[/math]
- [math]\displaystyle{ D }[/math] is closed under countable unions of pairwise disjoint sets: if [math]\displaystyle{ A_1, A_2, A_3, \ldots }[/math] is a sequence of pairwise disjoint sets in [math]\displaystyle{ D }[/math] (meaning that [math]\displaystyle{ A_i \cap A_j = \varnothing }[/math] for all [math]\displaystyle{ i \neq j }[/math]) then [math]\displaystyle{ \bigcup_{n=1}^\infty A_n \in D. }[/math]
- To be clear, this property also holds for finite sequences [math]\displaystyle{ A_1, \ldots, A_n }[/math] of pairwise disjoint sets (by letting [math]\displaystyle{ A_i := \varnothing }[/math] for all [math]\displaystyle{ i \gt n }[/math]).
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 as they are easier to verify.
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 [math]\displaystyle{ \mathcal{J} }[/math] of subsets of [math]\displaystyle{ \Omega, }[/math] there exists a unique Dynkin system denoted [math]\displaystyle{ D\{\mathcal{J}\} }[/math] which is minimal with respect to containing [math]\displaystyle{ \mathcal J. }[/math] That is, if [math]\displaystyle{ \tilde D }[/math] is any Dynkin system containing [math]\displaystyle{ \mathcal{J}, }[/math] then [math]\displaystyle{ D\{\mathcal{J}\} \subseteq \tilde{D}. }[/math] [math]\displaystyle{ D\{\mathcal{J}\} }[/math] is called the Dynkin system generated by [math]\displaystyle{ \mathcal{J}. }[/math] For instance, [math]\displaystyle{ D\{\varnothing\} = \{\varnothing, \Omega\}. }[/math] For another example, let [math]\displaystyle{ \Omega = \{1,2,3,4\} }[/math] and [math]\displaystyle{ \mathcal{J} = \{1\} }[/math]; then [math]\displaystyle{ D\{\mathcal{J}\} = \{\varnothing, \{1\}, \{2,3,4\}, \Omega\}. }[/math]
Sierpiński–Dynkin's π-λ theorem
Sierpiński-Dynkin's π-𝜆 theorem:[3] If [math]\displaystyle{ P }[/math] is a π-system and [math]\displaystyle{ D }[/math] is a Dynkin system with [math]\displaystyle{ P\subseteq D, }[/math] then [math]\displaystyle{ \sigma\{P\}\subseteq D. }[/math]
In other words, the 𝜎-algebra generated by [math]\displaystyle{ P }[/math] is contained in [math]\displaystyle{ D. }[/math] 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 [math]\displaystyle{ (\Omega, \mathcal{B}, \ell) }[/math] be the unit interval [0,1] with the Lebesgue measure on Borel sets. Let [math]\displaystyle{ m }[/math] be another measure on [math]\displaystyle{ \Omega }[/math] satisfying [math]\displaystyle{ m[(a, b)] = b - a, }[/math] and let [math]\displaystyle{ D }[/math] be the family of sets [math]\displaystyle{ S }[/math] such that [math]\displaystyle{ m[S] = \ell[S]. }[/math] Let [math]\displaystyle{ I := \{ (a, b), [a, b), (a, b], [a, b] : 0 \lt a \leq b \lt 1 \}, }[/math] and observe that [math]\displaystyle{ I }[/math] is closed under finite intersections, that [math]\displaystyle{ I \subseteq D, }[/math] and that [math]\displaystyle{ \mathcal{B} }[/math] is the 𝜎-algebra generated by [math]\displaystyle{ I. }[/math] It may be shown that [math]\displaystyle{ D }[/math] satisfies the above conditions for a Dynkin-system. From Sierpiński-Dynkin's π-𝜆 Theorem it follows that [math]\displaystyle{ D }[/math] in fact includes all of [math]\displaystyle{ \mathcal{B} }[/math], which is equivalent to showing that the Lebesgue measure is unique on [math]\displaystyle{ \mathcal{B} }[/math].
Application to probability distributions
See also
- Algebra of sets – Identities and relationships involving sets
- δ-ring – Ring closed under countable intersections
- Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
- π-system – Family of sets closed under intersection
- Ring of sets – Family closed under unions and relative complements
- Σ-algebra – Algebraic structure of set algebra
- 𝜎-ideal – Family closed under subsets and countable unions
- 𝜎-ring – Ring closed under countable unions
Notes
- ↑ A sequence of sets [math]\displaystyle{ A_1, A_2, A_3, \ldots }[/math] is called increasing if [math]\displaystyle{ A_n \subseteq A_{n+1} }[/math] for all [math]\displaystyle{ n \geq 1. }[/math]
Proofs
- ↑ Assume [math]\displaystyle{ \mathcal{D} }[/math] satisfies (1), (2), and (3). Proof of (5) :Property (5) follows from (1) and (2) by using [math]\displaystyle{ B := \Omega. }[/math] The following lemma will be used to prove (6). Lemma: If [math]\displaystyle{ A, B \in \mathcal{D} }[/math] are disjoint then [math]\displaystyle{ A \cup B \in \mathcal{D}. }[/math] Proof of Lemma: [math]\displaystyle{ A \cap B = \varnothing }[/math] implies [math]\displaystyle{ B \subseteq \Omega \setminus A, }[/math] where [math]\displaystyle{ \Omega \setminus A \subseteq \Omega }[/math] by (5). Now (2) implies that [math]\displaystyle{ \mathcal{D} }[/math] contains [math]\displaystyle{ (\Omega \setminus A) \setminus B = \Omega \setminus (A \cup B) }[/math] so that (5) guarantees that [math]\displaystyle{ A \cup B \in \mathcal{D}, }[/math] which proves the lemma. Proof of (6) Assume that [math]\displaystyle{ A_1, A_2, A_3, \ldots }[/math] are pairwise disjoint sets in [math]\displaystyle{ \mathcal{D}. }[/math] For every integer [math]\displaystyle{ n \gt 0, }[/math] the lemma implies that [math]\displaystyle{ D_n := A_1 \cup \cdots \cup A_n \in \mathcal{D} }[/math] where because [math]\displaystyle{ D_1 \subseteq D_2 \subseteq D_3 \subseteq \cdots }[/math] is increasing, (3) guarantees that [math]\displaystyle{ \mathcal{D} }[/math] contains their union [math]\displaystyle{ D_1 \cup D_2 \cup \cdots = A_1 \cup A_2 \cup \cdots, }[/math] as desired. [math]\displaystyle{ \blacksquare }[/math]
- ↑ Assume [math]\displaystyle{ \mathcal{D} }[/math] satisfies (4), (5), and (6). proof of (2): If [math]\displaystyle{ A, B \in \mathcal{D} }[/math] satisfy [math]\displaystyle{ A \subseteq B }[/math] then (5) implies [math]\displaystyle{ \Omega \setminus B \in \mathcal{D} }[/math] and since [math]\displaystyle{ (\Omega \setminus B) \cap A = \varnothing, }[/math] (6) implies that [math]\displaystyle{ \mathcal{D} }[/math] contains [math]\displaystyle{ (\Omega \setminus B) \cup A = \Omega \setminus (B \setminus A) }[/math] so that finally (4) guarantees that [math]\displaystyle{ \Omega \setminus (\Omega \setminus (B \setminus A)) = B \setminus A }[/math] is in [math]\displaystyle{ \mathcal{D}. }[/math] Proof of (3): Assume [math]\displaystyle{ A_1 \subseteq A_2 \subseteq \cdots }[/math] is an increasing sequence of subsets in [math]\displaystyle{ \mathcal{D}, }[/math] let [math]\displaystyle{ D_1 = A_1, }[/math] and let [math]\displaystyle{ D_i = A_i \setminus A_{i-1} }[/math] for every [math]\displaystyle{ i \gt 1, }[/math] where (2) guarantees that [math]\displaystyle{ D_2, D_3, \ldots }[/math] all belong to [math]\displaystyle{ \mathcal{D}. }[/math] Since [math]\displaystyle{ D_1, D_2, D_3, \ldots }[/math] are pairwise disjoint, (6) guarantees that their union [math]\displaystyle{ D_1 \cup D_2 \cup D_3 \cup \cdots = A_1 \cup A_2 \cup A_3 \cup \cdots }[/math] belongs to [math]\displaystyle{ \mathcal{D}, }[/math] which proves (3).[math]\displaystyle{ \blacksquare }[/math]
- ↑ Dynkin, E., "Foundations of the Theory of Markov Processes", Moscow, 1959
- ↑ Aliprantis, Charalambos; Border, Kim C. (2006). Infinite Dimensional Analysis: a Hitchhiker's Guide (Third ed.). Springer. https://books.google.com/books?id=4vyXtR3vUhoC&pg=PA135. Retrieved August 23, 2010.
- ↑ Sengupta. "Lectures on measure theory lecture 6: The Dynkin π − λ Theorem". https://www.math.lsu.edu/~sengupta/7360f09/DynkinPiLambda.pdf.
References
- Gut, Allan (2005). Probability: A Graduate Course. New York: Springer. doi:10.1007/b138932. ISBN 0-387-22833-0.
- Billingsley, Patrick (1995). Probability and Measure. New York: John Wiley & Sons, Inc.. ISBN 0-471-00710-2.
- Williams, David (2007). Probability with Martingales. Cambridge University Press. p. 193. ISBN 0-521-40605-6. https://books.google.com/books?id=e9saZ0YSi-AC&pg=PA193.
 |  |
