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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[edit]

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

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

Properties[edit]

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[edit]

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[edit]

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 ${\mathcal {R}}.$

References[edit]

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

Families ${\mathcal {F}}$ of sets over $\Omega$
Is necessarily true of ${\mathcal {F}}\colon$ or, is ${\mathcal {F}}$ closed under:	Directed by $\,\supseteq$	$A\cap B$	$A\cup B$	$B\setminus A$	$\Omega \setminus A$	$A_{1}\cap A_{2}\cap \cdots$	$A_{1}\cup A_{2}\cup \cdots$	$\Omega \in {\mathcal {F}}$	$\varnothing \in {\mathcal {F}}$	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
Dual ideal
Filter				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Prefilter (Filter base)				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Filter subbase				Never	Never				$\varnothing \not \in {\mathcal {F}}$
Open Topology							(even arbitrary $\cup$ )			Never
Closed Topology						(even arbitrary $\cap$ )				Never
Is necessarily true of ${\mathcal {F}}\colon$ or, is ${\mathcal {F}}$ 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 ${\mathcal {F}}.$ A *semialgebra* is a semiring where every complement $\Omega \setminus A$ is equal to a finite disjoint union of sets in ${\mathcal {F}}.$ $A,B,A_{1},A_{2},\ldots$ are arbitrary elements of ${\mathcal {F}}$ and it is assumed that ${\mathcal {F}}\neq \varnothing .$

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Formal definition[edit]

Properties[edit]

Similar concepts[edit]

Uses[edit]

See also[edit]

References[edit]

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