Sigma-ring

Template:Short description In mathematics, a nonempty collection of sets is called a Template:Sigma-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation.

Formal definition

Let $ℛ$ be a nonempty collection of sets. Then $ℛ$ is a Template:Sigma-ring if:

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

Properties

These two properties imply: $⋂_{n = 1}^{\infty} A_{n} \in ℛ$ whenever $A_{1}, A_{2}, \dots$ are elements of $ℛ .$

This is because $⋂_{n = 1}^{\infty} A_{n} = A_{1} ∖ ⋃_{n = 2}^{\infty} (A_{1} ∖ A_{n}) .$

Every Template:Sigma-ring is a δ-ring but there exist δ-rings that are not Template:Sigma-rings.

Similar concepts

If the first property is weakened to closure under finite union (that is, $A \cup B \in ℛ$ whenever $A, B \in ℛ$ ) but not countable union, then $ℛ$ is a ring but not a Template:Sigma-ring.

Uses

Template:Sigma-rings can be used instead of [[Sigma-algebra|Template:Sigma-fields]] (Template:Sigma-algebras) in the development of measure and integration theory, if one does not wish to require that the universal set be measurable. Every Template:Sigma-field is also a Template:Sigma-ring, but a Template:Sigma-ring need not be a Template:Sigma-field.

A Template:Sigma-ring $ℛ$ that is a collection of subsets of $X$ induces a [[Sigma-algebra|Template:Sigma-field]] for $X .$ Define $𝒜 = {E \subseteq X : E \in ℛ or E^{c} \in ℛ} .$ Then $𝒜$ is a Template:Sigma-field over the set $X$ - to check closure under countable union, recall a $σ$ -ring is closed under countable intersections. In fact $𝒜$ is the minimal Template:Sigma-field containing $ℛ$ since it must be contained in every Template:Sigma-field containing $ℛ .$

References

Template:Reflist

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

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Additionally, a Template:Em is a [[pi-system\|Template:Pi-system]] where every complement $B ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ A Template:Em is a semiring where every complement $Ω ∖ A$ is equal to a finite disjoint union of sets in $ℱ .$ $A, B, A_{1}, A_{2}, \dots$ are arbitrary elements of $ℱ$ and it is assumed that $ℱ \neq \emptyset .$

Sigma-ring

Contents

Formal definition

Properties

Similar concepts

Uses

See also

References

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Sigma-ring

Formal definition

Properties

Similar concepts

Uses

See also

References

Navigation menu

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