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 – 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