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

1. $\bigcup _{n=1}^{\infty }A_{n}\in {\mathcal {R}}$ if $A_{n}\in {\mathcal {R}}$ for all $n\in \mathbb {N}$
2. $A\setminus B\in {\mathcal {R}}$ if $A,B\in {\mathcal {R}}$

Properties

These two properties imply:

$\bigcap _{n=1}^{\infty }A_{n}\in {\mathcal {R}}$ if $A_{1},A_{2},\ldots$ are elements of </math>\mathcal{R}.</math>

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

If the first property is weakened to closure under finite union (i.e., $A\cup B\in {\mathcal {R}}$ whenever $A,B\in {\mathcal {R}}$) but not countable union, then ${\mathcal {R}}$ is a ring but not a σ-ring.

Uses

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 ${\mathcal {A}}=\{A\cup B^{c}|A,B\in {\mathcal {R}}\}$. 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 ${\mathcal {R}}$.

See also

• Algebra of sets
• δ-ring
• Field of sets
• λ-system (Dynkin system)
• π-system
• Ring of sets
• σ-algebra

References

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