Dynkin system

A Dynkin system, named after Eugene Dynkin, is a collection of subsets of another universal set

Ω

{\displaystyle \Omega }

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. 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

Ω

{\displaystyle \Omega }

be a set, and let

D

{\displaystyle D}

be a collection of subsets of

Ω

{\displaystyle \Omega }

(that is,

D

{\displaystyle D}

is a subset of the power set of

Ω

{\displaystyle \Omega }

). Then

D

{\displaystyle D}

is a Dynkin system if

Ω

∈

D

;

{\displaystyle \Omega \in D;}

D

{\displaystyle D}

is closed under complements of subsets in supersets: if

A

,

B

∈

D

{\displaystyle A,B\in D}

and

A

⊆

B

,

{\displaystyle A\subseteq B,}

then

B

∖

A

∈

D

;

{\displaystyle B\setminus A\in D;}

D

{\displaystyle D}

is closed under countable increasing unions: if

A

1

⊆

A

2

⊆

A

3

⊆

⋯

{\displaystyle A_{1}\subseteq A_{2}\subseteq A_{3}\subseteq \cdots }

is an increasing sequence of sets in

D

{\displaystyle D}

then

⋃

n

=

1

∞

A

n

∈

D

.

{\displaystyle \bigcup _{n=1}^{\infty }A_{n}\in D.}

It is easy to check that any Dynkin system

D

{\displaystyle D}

satisfies:

∅

∈

D

;

{\displaystyle \varnothing \in D;}

D

{\displaystyle D}

is closed under complements in

Ω

{\displaystyle \Omega }

: if

A

∈

D

,

{\textstyle A\in D,}

then

Ω

∖

A

∈

D

;

{\displaystyle \Omega \setminus A\in D;}

Taking

A

:=

Ω

{\displaystyle A:=\Omega }

shows that

∅

∈

D

.

{\displaystyle \varnothing \in D.}

D

{\displaystyle D}

is closed under countable unions of pairwise disjoint sets: if

A

1

,

A

2

,

A

3

,

…

{\displaystyle A_{1},A_{2},A_{3},\ldots }

is a sequence of pairwise disjoint sets in

D

{\displaystyle D}

(meaning that

A

i

∩

A

j

=

∅

{\displaystyle A_{i}\cap A_{j}=\varnothing }

for all

i

≠

j

{\displaystyle i\neq j}

) then

⋃

n

=

1

∞

A

n

∈

D

.

{\displaystyle \bigcup _{n=1}^{\infty }A_{n}\in D.}

To be clear, this property also holds for finite sequences

A

1

,

…

,

A

n

{\displaystyle A_{1},\ldots ,A_{n}}

of pairwise disjoint sets (by letting

A

i

:=

∅

{\displaystyle A_{i}:=\varnothing }

for all

i

>

n

{\displaystyle i>n}

).

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