---
tag: PFMGZ7988V
kind: concept
name: Algebra of sets
msc: [28A05]
specializes: [KWE1DBMN9B]
sources:
  - url: https://en.wikipedia.org/wiki/Field_of_sets
same_as:
  wikidata: Q246506
  wikipedia: Field of sets
---

## Summary

An algebra of sets on a set $X$ is a family of subsets of $X$ that contains $X$.
It is closed under complements and finite unions.
It is also called a field of sets.
Algebras of sets are the natural domain of finitely additive measures.
They are also the starting point of Carathéodory's construction of measures.

## Explanations

### L1 human

An algebra of sets is a family of subsets that is stable under the basic set operations done finitely many times.
If two sets are in the family, then so are their union, their intersection and their complements.
For example, on the natural numbers, the finite sets together with the sets whose complement is finite form an algebra.
A σ-algebra asks more: it must also be stable under countable unions.