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Delta-Ring


Given a set X, let F be a nonempty set of subsets of X. Then F is a ring if, for every pair of sets in F, the intersection, union, and set difference is also in F. F is called a delta-ring if F is a ring and, for any countable collection of sets A_n in F, the intersection  intersection A_n is also in F. A delta-ring F is sigma-finite if X is the union of a countable collection of sets in F.

Given a collection S of subsets of X, the delta-ring generated by S can be defined as the intersection of all delta-rings containing S. For example, the collection of bounded real Borel sets is a delta-ring. More generally, if X is a Hausdorff topological space, then the collection of Borel sets with compact closure is a delta-ring.

Unbounded (complex) measures are defined on delta-rings. If F is a sigma-algebra, it is a delta-ring, and if it is a delta-ring, it is a ring.

A ring of sets in X is also a ring in algebraic sense, if addition is defined as

 A+B=(A union B)\(A intersection B)=(A\B) union (B\A)

and multiplication as

 A*B=A intersection B.

See also

Ring, Sigma-Algebra

This entry contributed by Allan Cortzen

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Cite this as:

Cortzen, Allan. "Delta-Ring." From MathWorld--A Wolfram Web Resource, created by Eric W. Weisstein. https://mathworld.wolfram.com/Delta-Ring.html

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