# 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.