Sigma-ring

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In mathematics, a nonempty collection of sets \mathcal{R} is called a σ-ring (pronounced sigma-ring) if it is closed under countable union and relative complementation:

  1. \bigcup_{n=1}^{\infty} A_{n} \in \mathcal{R} if A_{n} \in \mathcal{R} for all n \in \mathbb{N}
  2. A \smallsetminus B \in \mathcal{R} if A, B \in \mathcal{R}

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 \smallsetminus \cup_{n=1}^{\infty}(A_1 \smallsetminus A_n).

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.

σ-rings can be used instead of σ-fields 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 induces a σ-field. If \mathcal{R} is a σ-ring over the set X, then 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}. We see that \mathcal{A} is a σ-field over the set X.

See also[edit]

References[edit]

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