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

  1. A \cup B \in \mathcal{R} if A, B \in \mathcal{R}
  2. A - B \in \mathcal{R} if A, B \in \mathcal{R}
  3. \bigcap_{n=1}^{\infty} A_{n} \in \mathcal{R} if A_{n} \in \mathcal{R} for all n \in \mathbb{N}

If only the first two properties are satisfied, then \mathcal{R} is a ring but not a δ-ring. Every σ-ring is a δ-ring, but not every δ-ring is a σ-ring.

δ-rings can be used instead of σ-fields in the development of measure theory if one does not wish to allow sets of infinite measure.

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