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.

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 ${\displaystyle A,B\in {\mathcal {R}}}$

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.

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.

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 ${\displaystyle {\mathcal {R}}.}$

References

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
Filter
Proper filter				Never	Never				Never
Prefilter (filter base)
Filter subbase
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 ${\displaystyle {\mathcal {F}}\neq \varnothing .}$