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{{Short description|Family closed under complements and countable disjoint unions}}
A '''Dynkin system''',<ref> Dynkin, E., "Foundations of the Theory of Markov Processes", Moscow, 1959 </ref> named after [[Eugene Dynkin]], is a [[Class (set theory)|collection]] of [[subset]]s of another universal [[Set (mathematics)|set]] <math>\Omega</math> satisfying a set of [[axiom]]s weaker than those of [[Sigma algebra|{{sigma}}-algebra]].  Dynkin systems are sometimes referred to as '''{{lambda}}-systems''' (Dynkin himself used this term) or '''d-system'''.<ref>{{cite book|title=Infinite Dimensional Analysis: a Hitchhiker's Guide|edition=Third|first1=Charalambos|last1=Aliprantis|first2=Kim C.|last2=Border |publisher=Springer|year=2006|isbn=978-3-540-29587-7 |url=https://books.google.com/books?id=4vyXtR3vUhoC&pg=PA135|accessdate=August 23, 2010}}</ref> These set families have applications in [[measure theory]] and [[probability]].

A major application of {{lambda}}-systems is the {{pi}}-{{lambda}} theorem, see below.

==Definition==

Let <math>\Omega</math> be a [[nonempty]] set, and let <math>D</math> be a [[Family of sets|collection of subsets]] of <math>\Omega</math> (that is, <math>D</math> is a subset of the [[power set]] of <math>\Omega</math>). Then <math>D</math> is a Dynkin system if
# <math>\Omega \in D;</math>
# <math>D</math> is closed under [[Complement (set theory)|complements]] of subsets in supersets: if <math>A, B \in D</math> and <math>A \subseteq B,</math> then <math>B \setminus A \in D;</math>
# <math>D</math> is closed under countable increasing [[Union (set theory)|union]]s: if <math>A_1 \subseteq A_2 \subseteq A_3 \subseteq \cdots</math> is an increasing sequence<ref group=note>A sequence of sets <math>A_1, A_2, A_3, \ldots</math> is called {{em|increasing}} if <math>A_n \subseteq A_{n+1}</math> for all <math>n \geq 1.</math></ref> of sets in <math>D</math> then <math>\bigcup_{n=1}^\infty A_n \in D.</math>

It is easy to check<ref group=note name=DynkinClosedUnderDisjointU /> that any Dynkin system <math>D</math> satisfies:
<ol start=4>
<li><math>\varnothing \in D;</math></li>
<li><math>D</math> is closed under complements in <math>\Omega</math>: if <math display="inline">A \in D,</math> then <math>\Omega \setminus A \in D;</math>
* Taking <math>A := \Omega</math> shows that <math>\varnothing \in D.</math></li>
<li><math>D</math> is closed under countable unions of [[pairwise disjoint]] sets: if <math>A_1, A_2, A_3, \ldots</math> is a sequence of [[pairwise disjoint]] sets in <math>D</math> (meaning that <math>A_i \cap A_j = \varnothing</math> for all <math>i \neq j</math>) then <math>\bigcup_{n=1}^\infty A_n \in D.</math>
* To be clear, this property also holds for finite sequences <math>A_1, \ldots, A_n</math> of pairwise disjoint sets (by letting <math>A_i := \varnothing</math> for all <math>i > n</math>).</li>
</ol>

Conversely, it is easy to check that a family of sets that satisfy conditions 4-6 is a Dynkin class.<ref group=note name=DisjointUGlobalComplementGiveDynkin />
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 [[Pi system|{{pi}}-system]] (that is, closed under finite intersections) is a [[Sigma algebra|{{sigma}}-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>\mathcal{J}</math> of subsets of <math>\Omega,</math> there exists a unique Dynkin system denoted <math>D\{\mathcal{J}\}</math> which is minimal with respect to containing <math>\mathcal J.</math> That is, if <math>\tilde D</math> is any Dynkin system containing <math>\mathcal{J},</math> then <math>D\{\mathcal{J}\} \subseteq \tilde{D}.</math> <math>D\{\mathcal{J}\}</math> is called the {{em|Dynkin system generated by <math>\mathcal{J}.</math>}} 
For instance, <math>D\{\varnothing\} = \{\varnothing, \Omega\}.</math> 
For another example, let <math>\Omega = \{1,2,3,4\}</math> and <math>\mathcal{J} = \{1\}</math>; then <math>D\{\mathcal{J}\} = \{\varnothing, \{1\}, \{2,3,4\}, \Omega\}.</math>

==Sierpiński–Dynkin's π-λ theorem{{anchor|Sierpiński-Dynkin's π-λ theorem|Dynkin's π-λ theorem|π-λ theorem|Pi-lambda theorem}}==

Sierpiński-Dynkin's {{pi}}-{{lambda}} theorem:<ref>{{cite web |last1=Sengupta |title=Lectures on measure theory lecture 6: The Dynkin π − λ Theorem  |url=https://www.math.lsu.edu/~sengupta/7360f09/DynkinPiLambda.pdf |website=Math.lsu |access-date=3 January 2023}}</ref>
If <math>P</math> is a [[Pi-system|{{pi}}-system]] and <math>D</math> is a Dynkin system with <math>P\subseteq D,</math> then <math>\sigma\{P\}\subseteq D.</math> 

In other words, the {{sigma}}-algebra generated by <math>P</math> is contained in <math>D.</math> Thus a Dynkin system contains a {{pi}}-system if and only if it contains the {{sigma}}-algebra generated by that {{pi}}-system. 

One application of Sierpiński-Dynkin's {{pi}}-{{lambda}} theorem is the uniqueness of a measure that evaluates the length of an interval (known as the [[Lebesgue measure]]):

Let <math>(\Omega, \mathcal{B}, \ell)</math> be the [[unit interval]] [0,1] with the Lebesgue measure on [[Borel sets]].  Let <math>m</math> be another [[Measure (mathematics)|measure]] on <math>\Omega</math> satisfying <math>m[(a, b)] = b - a,</math> and let <math>D</math> be the family of sets <math>S</math> such that <math>m[S] = \ell[S].</math>  Let <math>I := \{ (a, b), [a, b), (a, b], [a, b] : 0 < a \leq b < 1 \},</math> and observe that <math>I</math> is closed under finite intersections, that <math>I \subseteq D,</math> and that <math>\mathcal{B}</math> is the {{sigma}}-algebra generated by <math>I.</math>  It may be shown that <math>D</math> satisfies the above conditions for a Dynkin-system.  From Sierpiński-Dynkin's {{pi}}-{{lambda}} Theorem it follows that <math>D</math> in fact includes all of <math>\mathcal{B}</math>, which is equivalent to showing that the Lebesgue measure is unique on <math>\mathcal{B}</math>.

===Application to probability distributions===

{{Transcluded section|pi system}}
{{:Pi system}}

==See also==

* {{annotated link|Algebra of sets}}
* {{annotated link|Delta-ring|{{delta}}-ring}}
* {{annotated link|Field of sets}}
* {{annotated link|Monotone class}}
* {{annotated link|Pi-system|{{pi}}-system}}
* {{annotated link|Ring of sets}}
* {{annotated link|σ-algebra}}
* {{annotated link|Sigma-ideal|{{sigma}}-ideal}}
* {{annotated link|Sigma-ring|{{sigma}}-ring}}

==Notes==

{{reflist|group=note|refs=
<ref group=note name=DynkinClosedUnderDisjointU>Assume <math>\mathcal{D}</math> satisfies (1), (2), and (3). ''Proof of (5)'': Property (5) follows from (1) and (2) by using <math>B := \Omega.</math> The following lemma will be used to prove (6). ''Lemma'': If <math>A, B \in \mathcal{D}</math> are disjoint then <math>A \cup B \in \mathcal{D}.</math> 
''Proof of Lemma'': <math>A \cap B = \varnothing</math> implies <math>B \subseteq \Omega \setminus A,</math> where <math>\Omega \setminus A \subseteq \Omega</math> by (5). Now (2) implies that <math>\mathcal{D}</math> contains <math>(\Omega \setminus A) \setminus B = \Omega \setminus (A \cup B)</math> so that (5) guarantees that <math>A \cup B \in \mathcal{D},</math> which proves the lemma. ''Proof of (6)'' Assume that <math>A_1, A_2, A_3, \ldots</math> are pairwise disjoint sets in <math>\mathcal{D}.</math> For every integer <math>n > 0,</math> the lemma implies that <math>D_n := A_1 \cup \cdots \cup A_n \in \mathcal{D}</math> where because <math>D_1 \subseteq D_2 \subseteq D_3 \subseteq \cdots</math> is increasing, (3) guarantees that <math>\mathcal{D}</math> contains their union <math>D_1 \cup D_2 \cup \cdots = A_1 \cup A_2 \cup \cdots,</math> as desired. <math>\blacksquare</math>
</ref>
<ref group=note name=DisjointUGlobalComplementGiveDynkin>Assume <math>\mathcal{D}</math> satisfies (4), (5), and (6). ''Proof of (2)'': If <math>A, B \in \mathcal{D}</math> satisfy <math>A \subseteq B</math> then (5) implies <math>\Omega \setminus B \in \mathcal{D}</math> and since <math>(\Omega \setminus B) \cap A = \varnothing,</math> (6) implies that <math>\mathcal{D}</math> contains <math>(\Omega \setminus B) \cup A = \Omega \setminus (B \setminus A)</math> so that finally (4) guarantees that <math>\Omega \setminus (\Omega \setminus (B \setminus A)) = B \setminus A</math> is in <math>\mathcal{D}.</math> ''Proof of (3)'': Assume <math>A_1 \subseteq A_2 \subseteq \cdots</math> is an increasing sequence of subsets in <math>\mathcal{D},</math> let <math>D_1 = A_1,</math> and let <math>D_i = A_i \setminus A_{i-1}</math> for every <math>i > 1,</math> where (2) guarantees that <math>D_2, D_3, \ldots</math> all belong to <math>\mathcal{D}.</math> Since <math>D_1, D_2, D_3, \ldots</math> are pairwise disjoint, (6) guarantees that their union <math>D_1 \cup D_2 \cup D_3 \cup \cdots = A_1 \cup A_2 \cup A_3 \cup \cdots</math> belongs to <math>\mathcal{D},</math> which proves (3).<math>\blacksquare</math>
</ref>
}}

==References==

{{reflist}}

==Further reading==

* {{cite book
  | last       = Gut
  | first      = Allan
  | title      = Probability: A Graduate Course
  | series = Springer Texts in Statistics
 | publisher  = Springer
  | year       = 2005
  | location   = New York
  | doi        = 10.1007/b138932
  | isbn       = 0-387-22833-0
}}
* {{cite book
  | last       = Billingsley
  | first      = Patrick
  | author-link= Patrick Billingsley
  | title      = Probability and Measure
  | publisher  = John Wiley & Sons, Inc.
  | year       = 1995
  | location   = New York
  | isbn       = 0-471-00710-2
}}
* {{cite book | first=David |last=Williams | author-link = David Williams (mathematician) | year=2007 | title=Probability with Martingales | publisher=Cambridge University Press | isbn=978-0-521-40605-5 | page=193 |url=https://books.google.com/books?id=e9saZ0YSi-AC&pg=PA193 }}

{{PlanetMath attribution|id=2024|title=Dynkin system}}

{{Families of sets}}
{{Measure theory}}

[[Category:Families of sets]]
[[Category:Lemmas]]
[[Category:Probability theory]]