setalgebra

A set algebra, or algebra of sets, on a universal set X is a nonempty collection A of subsets of X that contains X and is closed under taking complements with respect to X and under finite unions. Equivalently, if E and F are in A, then E ∪ F and E^c (the complement of E in X) are in A. From these properties it follows that ∅ = X \ X also lies in A, and A is closed under finite intersections as well.

In structural terms, a set algebra forms a Boolean algebra under the operations of union, intersection, and complement, with X serving as the greatest element and ∅ as the least element. This perspective emphasizes the logical interpretation of events and their combinations in measure theory and probability.

Examples range from the simplest to the more structured. The power set P(X) is a set algebra on X. More generally, if X is partitioned into a finite or countable family of disjoint blocks, the collection of all unions of these blocks is a set algebra. When X is finite, every set algebra on X is automatically a sigma-algebra, since any countable union reduces to a finite union.

A sigma-algebra is a special kind of set algebra that is closed under countable unions (and thus under complements). Therefore, set algebras provide a finite-closure framework for the events in a probability space or a measure space, while sigma-algebras extend this to countable operations. The notion of a set algebra is closely related to Boolean algebra and to the concept of generating algebras from partitions.

See also: Boolean algebra, sigma-algebra, measurable space, ring of sets, algebra of sets generation.