Dynkin system

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, and $\mathcal{D}$ is closed under countable unions of pairwise disjoint members. Every σ-algebra is a Dynkin system. A Dynkin system that is also a π-system is a σ-algebra. Dynkin systems are also called λ-systems.