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:

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

Properties

From these two properties we immediately see that

\bigcap _{{n=1}}^{{\infty }}A_{n}\in {\mathcal {R}}

A_{{n}}\in {\mathcal {R}}

for all

n\in \mathbb {N}

This is simply because $\cap _{{n=1}}^{\infty }A_{n}=A_{1}\setminus \cup _{{n=1}}^{{\infty }}(A_{1}\setminus A_{n})$ .

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 ${\mathcal {A}}$ to be the collection of all subsets of $X$ that are elements of ${\mathcal {R}}$ or whose complements are elements of ${\mathcal {R}}$ . Then ${\mathcal {A}}$ is a σ-field over the set $X$ . 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

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

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