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In mathematics, a nonempty collection of sets is called a σ-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.

Formal definition[edit]

Let be a nonempty collection of sets. Then is a σ-ring if:

  1. if for all
  2. if


From these two properties we immediately see that

if for all

This is simply because .

Similar concepts[edit]

If the first property is weakened to closure under finite union (i.e., whenever ) but not countable union, then is a ring but not a σ-ring.


σ-rings can be used instead of σ-fields (σ-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every σ-field is also a σ-ring, but a σ-ring need not be a σ-field.

A σ-ring that is a collection of subsets of induces a σ-field for . Define to be the collection of all subsets of that are elements of or whose complements are elements of . Then is a σ-field over the set . In fact is the minimal σ-field containing since it must be contained in every σ-field containing .

See also[edit]


  • Walter Rudin, 1976. Principles of Mathematical Analysis, 3rd. ed. McGraw-Hill. Final chapter uses σ-rings in development of Lebesgue theory.