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Sigma-ring

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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.

Contents

  • Formal definition 1
  • Properties 2
  • Similar concepts 3
  • Uses 4
  • See also 5
  • References 6

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

From these two properties we immediately see that

\bigcap_{n=1}^{\infty} A_n \in \mathcal{R} if 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}.

See also

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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