---
tag: 69460Z6DMW
kind: definition
name: Dynkin system
msc: [28A05, 60A10]
defines: V5M2CKB591
uses: [0SJ3RX9TXA, 07KVB4JSC7, 0B1FSS1VZW, 04WSW8T3SY, 4W9NM803BB]
sources:
  - url: https://en.wikipedia.org/wiki/Dynkin_system
---

## Definition

Let $X$ be a [[set|0SJ3RX9TXA]].
A *Dynkin system* on $X$ is a family $\mathcal{D}$ of subsets of $X$ such that:

1. $X \in \mathcal{D}$.
2. If $A, B \in \mathcal{D}$ and $A \subseteq B$, then $B \setminus A \in \mathcal{D}$.
3. If $A_1 \subseteq A_2 \subseteq \cdots$ is an increasing sequence of members of $\mathcal{D}$, then the [[union|07KVB4JSC7]] $\bigcup_{n=1}^{\infty} A_n$ is in $\mathcal{D}$.

## Remarks

On $X = \set{1, 2, 3, 4}$, the family of subsets with an even number of elements is a Dynkin system.
Clause 1 holds since $X$ has $4$ elements.
Clause 2 holds since $B \setminus A$ has $\abs{B} - \abs{A}$ elements when $A \subseteq B$.
Clause 3 holds since an increasing sequence in a finite family is eventually constant.
On $X = \set{1, 2}$, the family $\set{\varnothing, \set{1}, X}$ is not a Dynkin system, since clause 2 fails: $X \setminus \set{1} = \set{2}$ is missing.
An equivalent definition: $X \in \mathcal{D}$, $\mathcal{D}$ is closed under [[complements|0B1FSS1VZW]], and $\mathcal{D}$ is closed under countable unions of pairwise disjoint members.
Every [[σ-algebra|04WSW8T3SY]] is a Dynkin system.
A Dynkin system that is also a [[π-system|4W9NM803BB]] is a σ-algebra.
Dynkin systems are also called λ-systems.