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:

1. Closed under countable unions: $\bigcup _{n=1}^{\mathrm{\infty }}{A}_n\in ℛ$ if $A_n\in ℛ$ for all $n\in \mathbb{N}$
2. Closed under relative complementation: $A\setminus B\in ℛ$ if $A,B\in ℛ$

Properties

These two properties imply: $$\bigcap _{n=1}^{\mathrm{\infty }}{A}_n\in ℛ$$ whenever $A_1,A_2,\dots$ are elements of $ℛ.$

This is because $$\bigcap _{n=1}^{\mathrm{\infty }}{A}_n=A_1\setminus \bigcup _{n=2}^{\mathrm{\infty }}(A_1\setminus A_n).$$

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 $𝒜=\{E\subseteq X:E\in ℛ\text{or}{E}^c\in ℛ\}.$ Then $𝒜$ 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 $𝒜$ is the minimal 𝜎-field containing $ℛ$ since it must be contained in every 𝜎-field containing $ℛ.$

See also

• δ-ring – Ring closed under countable intersections
• Field of sets – Algebraic concept in measure theory, also referred to as an algebra of sets
• Join (sigma algebra) – Algebraic structure of set algebraPages displaying short descriptions of redirect targets
• 𝜆-system (Dynkin system) – Family closed under complements and countable disjoint unions
• Measurable function – Kind of mathematical function
• Monotone class – Measure theory and probability theoremPages displaying short descriptions of redirect targets
• π-system – Family of sets closed under intersection
• Ring of sets – Family closed under unions and relative complements
• Sample space – Set of all possible outcomes or results of a statistical trial or experiment
• 𝜎 additivity – Mapping function
• σ-algebra – Algebraic structure of set algebra
• 𝜎-ideal – Family closed under subsets and countable unions

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 $\Omega$
Is necessarily true of $ℱ:$ or, is $ℱ$ closed under:	Directed by $\supseteq$	$A\cap B$	$A\cup B$	$B\setminus A$	$\Omega \setminus A$	$A_1\cap A_2\cap \cdots$	$A_1\cup A_2\cup \cdots$	$\Omega \in ℱ$	$\varnothing \in ℱ$	F.I.P.
π-system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if $A_i\searrow$	only if $A_i\nearrow$
𝜆-system (Dynkin System)				only if $A\subseteq B$			only if $A_i\nearrow$ 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							File:Green check.svg (even arbitrary $\cup$)			Never
Closed Topology						File:Green check.svg (even arbitrary $\cap$)				Never
Is necessarily true of $ℱ:$ or, is $ℱ$ closed under:	directed downward	finite intersections	finite unions	relative complements	complements in $\Omega$	countable intersections	countable unions	contains $\Omega$	contains $\varnothing$	Finite Intersection Property
Additionally, a semiring is a π-system where every complement $B\setminus A$ is equal to a finite disjoint union of sets in $ℱ.$ A semialgebra is a semiring where every complement $\Omega \setminus A$ 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 $ℱ\ne \varnothing .$