Dynkin system

A Dynkin system on a set is a collection of its subsets that contains the whole set and is closed under complements and countable disjoint unions. It is also called a bb-system.

Power set: The set of all subsets of a given set. A Dynkin system is a subcollection of this larger family.
Difference of nested sets: The elements in one set but not in a subset contained within it. Dynkin-system axioms imply closure under differences of nested members.
Probability measure: A countably additive measure assigning total mass one to a measurable space. Equality of probability measures can be extended from a generating pi-system using Dynkin-system arguments.
Pi-system: A collection of sets closed under finite intersections. The pi-lambda theorem combines this intersection closure with Dynkin-system closure.
Set complement: The elements of a universe that are not in a specified subset. Closure under complements is one of the defining Dynkin-system axioms.
Increasing union: The union of a sequence of sets, each contained in the next. Such unions belong to a Dynkin system after decomposing them into disjoint differences.
Uniqueness of measures: Theorems establishing when specified properties determine a measure uniquely. Dynkin systems help prove two measures agree everywhere from agreement on a suitable family.
Semiring of sets: A family of sets containing the empty set, closed under intersections, with differences expressible as finite disjoint unions. Semirings support measure construction through different closure properties.
Pairwise disjoint sets: A family of sets in which every two distinct members have empty intersection. The system's union axiom applies to countable families with this property.
Lambda system: A family of sets containing the universe and closed under complements and countable disjoint unions. This is another standard name for a Dynkin system.

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