Editing Field of sets (section)

{{redirect|Set algebra|the basic properties and laws of sets|Algebra of sets}}
{{Short description|Algebraic concept in measure theory, also referred to as an algebra of sets}}

In [[mathematics]], a '''field of sets''' is a [[mathematical structure]] consisting of a pair <math>( X, \mathcal{F} )</math> consisting of a [[Set (mathematics)|set]] <math>X</math> and a [[Family of sets|family]] <math>\mathcal{F}</math> of [[subset]]s of <math>X</math> called an '''algebra over <math>X</math>''' that contains the [[empty set]] as an element, and is closed under the operations of taking [[Complement (set theory)|complements]] in <math>X,</math> finite [[Union (set theory)|unions]], and finite [[Intersection (set theory)|intersections]].

Fields of sets should not be confused with [[Field (mathematics)|field]]s in [[ring theory]] nor with [[Field (physics)|fields in physics]]. Similarly the term "algebra over <math>X</math>" is used in the sense of a Boolean algebra and should not be confused with [[Algebra over a field|algebras over fields or rings]] in ring theory.

Fields of sets play an essential role in the [[representation theory]] of Boolean algebras. Every Boolean algebra can be represented as a field of sets.