Sigma-ring

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 definitionEdit

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

  1.   if   for all  
  2.   if  

PropertiesEdit

These two properties imply:

  if   are elements of  

This is because

 

Every σ-ring is a δ-ring but there exist δ-rings that are not σ-rings.

Similar conceptsEdit

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.

UsesEdit

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  . Then   is a σ-field over the set   - to check closure under countable union, recall a  -ring is closed under countable intersections. In fact   is the minimal σ-field containing   since it must be contained in every σ-field containing  .

See alsoEdit

ReferencesEdit

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