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:

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

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.