Set algebra

A set algebra on a set $X$ is a nonempty collection $\mathcal A \subseteq \mathcal P(X)$ such that if $A\in\mathcal A$ then $X\setminus A\in\mathcal A$, and if $A,B\in\mathcal A$ then $A\cup B\in\mathcal A$.

Here $\mathcal P(X)$ is the power set of the set $X$. Closure under complements and finite unions implies closure under finite intersections and finite set differences . A set algebra is the typical domain for a premeasure , and every sigma-algebra is a set algebra.

Examples:

• For any $X$, the full collection $\mathcal P(X)$ is a set algebra.
• The family of subsets of $\mathbb R$ that are finite unions of half-open intervals of the form $[a,b)$ is a set algebra.
• On an infinite set $X$, the collection of all finite subsets of $X$ together with all cofinite subsets of $X$ (those whose complement is finite) is a set algebra.