 Dynkin system

A Dynkin system on $\Omega$ contains $\Omega$, is closed under complements relative to $\Omega$, and is closed under countable disjoint unions. Unlike a sigma-algebra, it need not be closed under intersections of arbitrary members.

An equivalent axiom list is: contain $\Omega$, contain $B\setminus A$ whenever $A\subseteq B$ are members, and be closed under increasing countable unions. For the forward direction, $B\setminus A=(B^c\cup A{)}^c$, using a disjoint union. For the reverse direction, differences from $\Omega$ give complements; finite disjoint unions follow by two differences, and increasing partial unions give the countable disjoint-union axiom.

For example, on $\Omega =\{1,2,3,4\}$, take the empty set, $\Omega$, and every two-element subset. This is a Dynkin system: a disjoint union of two nonempty proper members is $\Omega$. It is not a sigma-algebra, since two intersecting pairs can have a singleton intersection. The Dynkin lemma says that adjoining intersection closure through a generating pi-system forces the full generated sigma-algebra.

A Dynkin system (also known as a π-system or a Dynkin π-system) is a collection of sets that satisfies certain properties, making it useful in measure theory and probability. Specifically, a collection \( \mathcal{D} \) of subsets of a given set \( X \) is called a Dynkin system if it satisfies the following properties: 1. **Contains the entire set**: \( X \in \mathcal{D} \).