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 definition

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

Closed under countable unions: $⋃_{n = 1}^{\infty} A_{n} \in ℛ$ if $A_{n} \in ℛ$ for all $n \in ℕ$
Closed under relative complementation: $A ∖ B \in ℛ$ if $A, B \in ℛ$

Properties

These two properties imply: $⋂_{n = 1}^{\infty} A_{n} \in ℛ$ whenever $A_{1}, A_{2}, \dots$ are elements of $ℛ .$

This is because $⋂_{n = 1}^{\infty} A_{n} = A_{1} ∖ ⋃_{n = 2}^{\infty} (A_{1} ∖ A_{n}) .$

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, $A \cup B \in ℛ$ whenever $A, B \in ℛ$ ) 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 $𝒜 = {E \subseteq X : E \in ℛ or E^{c} \in ℛ} .$ Then $𝒜$ is a 𝜎-field over the set $X$ - to check closure under countable union, recall a $σ$ -ring is closed under countable intersections. In fact $𝒜$ is the minimal 𝜎-field containing $ℛ$ since it must be contained in every 𝜎-field containing $ℛ .$

References

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

Families $ℱ$ of sets over $Ω$ v t e
Is necessarily true of $ℱ :$ or, is $ℱ$ closed under:	Directed by $\supseteq$	$A \cap B$	$A \cup B$	$B ∖ A$	$Ω ∖ A$	$A_{1} \cap A_{2} \cap \dots$	$A_{1} \cup A_{2} \cup \dots$	$Ω \in ℱ$	$\emptyset \in ℱ$	F.I.P.
$π$ -system
Semiring										Never
Semialgebra (semifield)										Never
Monotone class						only if $A_{i} ↘$	only if $A_{i} ↗$
𝜆-system (Dynkin system)				only if $A \subseteq B$			only if $A_{i} ↗$ or they are disjoint			Never
Ring (order theory)
Ring (measure theory)										Never
δ-ring										Never
𝜎-ring										Never
Algebra (field)										Never
𝜎-algebra (𝜎-field)										Never
Filter
Proper filter				Never	Never				Never
Prefilter (filter base)
Filter subbase
Open topology							(even arbitrary $\cup$ )			Never
Closed topology						(even arbitrary $\cap$ )				Never
Is necessarily true of $ℱ :$ or, is $ℱ$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in $Ω$	countable intersections	countable unions	contains $Ω$	contains $\emptyset$	Finite intersection property
Additionally, a *semiring* is a $π$ -system where every complement $B ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ A *semialgebra* is a semiring where every complement $Ω ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ $A, B, A_{1}, A_{2}, \dots$ are arbitrary elements of $ℱ$ and it is assumed that $ℱ \neq \emptyset .$