Sigma ring

Formal definition

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

1. ⋃ n = 1 ∞ A n ∈ R if A n ∈ R for all n ∈ N
2. A ∖ B ∈ R if A , B ∈ R

Properties

From these two properties we immediately see that

This is simply because ∩ n = 1 ∞ A n = A 1 ∖ ∪ n = 1 ∞ ( A 1 ∖ A n ) .

Similar concepts

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

Uses

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