Sigma-ring

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

Formal definition
Let $$\mathcal{R}$$ be a nonempty collection of sets. Then $$\mathcal{R}$$ is a 𝜎-ring if:
 * 1) Closed under countable unions: $$\bigcup_{n=1}^{\infty} A_{n} \in \mathcal{R}$$ if $$A_{n} \in \mathcal{R}$$ for all $$n \in \N$$
 * 2) Closed under relative complementation: $$A \setminus B \in \mathcal{R}$$ if $$A, B \in \mathcal{R}$$

Properties
These two properties imply: $$\bigcap_{n=1}^{\infty} A_n \in \mathcal{R}$$ whenever $$A_1, A_2, \ldots$$ are elements of $$\mathcal{R}.$$

This is because $$\bigcap_{n=1}^\infty A_n = A_1 \setminus \bigcup_{n=2}^{\infty}\left(A_1 \setminus A_n\right).$$

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

Similar concepts
If the first property is weakened to closure under finite union (that is, $$A \cup B \in \mathcal{R}$$ whenever $$A, B \in \mathcal{R}$$) but not countable union, then $$\mathcal{R}$$ 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 $$\mathcal{R}$$ that is a collection of subsets of $$X$$ induces a 𝜎-field for $$X.$$ Define $$\mathcal{A} = \{ E \subseteq X : E \in \mathcal{R} \ \text{or} \ E^c \in \mathcal{R} \}.$$ Then $$\mathcal{A}$$ is a 𝜎-field over the set $$X$$ - to check closure under countable union, recall a $$\sigma$$-ring is closed under countable intersections. In fact $$\mathcal{A}$$ is the minimal 𝜎-field containing $$\mathcal{R}$$ since it must be contained in every 𝜎-field containing $$\mathcal{R}.$$