Sigma-ring
In mathematics, a nonempty collection of sets is called a σ-ring if it is closed under countable union and relative complementation.Let be a nonempty collection of sets. Then is a σ-ring if:
- if for all
- if
Properties
From these two properties we immediately see that
This is simply because.If the first property is weakened to closure under finite union but not countable union, then is a ring but not a σ-ring.Uses
σ-rings can be used instead of σ-fields 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.
