Sigma-ring explained

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

Formal definition

Let

l{R} be a nonempty collection of sets. Then l{R} is a -ring if:

1. Closed under countable unions:

c up ^{inf ty} _{n =1} A _{n} \in l{R} if A _{n} \in l{R} for all n \in \N

1. Closed under relative complementation:

A \setminus B \in l{R} if A, B \in l{R}

Properties

These two properties imply:$$\backslash bigcap\_^\; A\_n\; \backslash in\; \backslash mathcal$$whenever

A _{1,} A _{2,} \ldots are elements of l{R}.

This is because$$\backslash bigcap\_^\backslash infty\; A\_n\; =\; A\_1\; \backslash setminus\; \backslash bigcup\_^\backslash left(A\_1\; \backslash setminus\; A\_n\backslash right).$$

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

Similar concepts

If the first property is weakened to closure under finite union (that is,

A \cup B \in l{R} whenever A, B \in l{R} ) but not countable union, then l{R} is a ring but not a -ring.

Uses

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

A -ring

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

References

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