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 ${\displaystyle {\mathcal {R}}}$  be a nonempty collection of sets. Then ${\displaystyle {\mathcal {R}}}$  is a 𝜎-ring if:

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

Properties

These two properties imply:

$${\displaystyle \bigcap _{n=1}^{\infty }A_{n}\in {\mathcal {R}}}$$

 

whenever ${\displaystyle A_{1},A_{2},\ldots }$  are elements of ${\displaystyle {\mathcal {R}}.}$ 

This is because

$${\displaystyle \bigcap _{n=1}^{\infty }A_{n}=A_{1}\setminus \bigcup _{n=2}^{\infty }\left(A_{1}\setminus A_{n}\right).}$$

 

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

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 – Function for which the preimage of a measurable set is measurable
• Monotone class – theoremPages displaying wikidata descriptions as a fallbackPages displaying short descriptions with no spaces
• 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 ${\displaystyle {\mathcal {F}}}$  of sets over ${\displaystyle \Omega }$  v t e
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$  or, is ${\displaystyle {\mathcal {F}}}$  closed under:	Directed by ${\displaystyle \,\supseteq }$	${\displaystyle A\cap B}$	${\displaystyle A\cup B}$	${\displaystyle B\setminus A}$	${\displaystyle \Omega \setminus A}$	${\displaystyle A_{1}\cap A_{2}\cap \cdots }$	${\displaystyle A_{1}\cup A_{2}\cup \cdots }$	${\displaystyle \Omega \in {\mathcal {F}}}$	${\displaystyle \varnothing \in {\mathcal {F}}}$	F.I.P.
π-system
Semiring										Never
Semialgebra (Semifield)										Never
Monotone class						only if ${\displaystyle A_{i}\searrow }$	only if ${\displaystyle A_{i}\nearrow }$
𝜆-system (Dynkin System)				only if ${\displaystyle A\subseteq B}$			only if ${\displaystyle A_{i}\nearrow }$  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
Dual ideal
Filter				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Prefilter (Filter base)				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Filter subbase				Never	Never				${\displaystyle \varnothing \not \in {\mathcal {F}}}$
Open Topology							(even arbitrary ${\displaystyle \cup }$ )			Never
Closed Topology						(even arbitrary ${\displaystyle \cap }$ )				Never
Is necessarily true of ${\displaystyle {\mathcal {F}}\colon }$  or, is ${\displaystyle {\mathcal {F}}}$  closed under:	directed downward	finite intersections	finite unions	relative complements	complements in ${\displaystyle \Omega }$	countable intersections	countable unions	contains ${\displaystyle \Omega }$	contains ${\displaystyle \varnothing }$	Finite Intersection Property
Additionally, a semiring is a π-system where every complement ${\displaystyle B\setminus A}$  is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$  A semialgebra is a semiring where every complement ${\displaystyle \Omega \setminus A}$  is equal to a finite disjoint union of sets in ${\displaystyle {\mathcal {F}}.}$  ${\displaystyle A,B,A_{1},A_{2},\ldots }$  are arbitrary elements of ${\displaystyle {\mathcal {F}}}$  and it is assumed that ${\displaystyle {\mathcal {F}}\neq \varnothing .}$